Continuity and Differentiability Class 12

Prove that the function $f(x)=5 x-3$ is continuous at $x=0$, at $x=-3$ and at $x=5$.

Examine the continuity of the function $f(x)=2 x^2-1$ at $x=3$.

Examine the following functions for continuity.
(a) $f(x)=x-5$

Examine the following functions for continuity.
(b) $f(x)=\frac{1}{x-5}, x \neq 5$

Examine the following functions for continuity.
(c) $f(x)=\frac{x^2-25}{x+5}, x \neq-5$

Examine the following functions for continuity.
(d) $f(x)=|x-5|$

Prove that the function $f(x)=x^n$ is continuous at $x=n$, where $n$ is a positive integer.

Is the function $f$ defined by

$$
f(x)= \begin{cases}x, & \text { if } x \leq 1 \\ 5, & \text { if } x>1\end{cases}
$$

continuous at $x=0$ ? At $x=1$ ? At $x=2$ ?
Find all points of discontinuity of $f$, where $f$ is defined by

$f(x)=\left\{\begin{array}{l}2 x+3, \text { if } x \leq 2 \\ 2 x-3, \text { if } x>2\end{array}\right.$

$f(x)=\left\{\begin{array}{cl}|x|+3, & \text { if } x \leq-3 \\ -2 x, & \text { if }-3<x<3 \\ 6 x+2, & \text { if } x \geq 3\end{array}\right.$

$f(x)=\left\{\begin{array}{cc}\frac{|x|}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.$

$f(x)= \begin{cases}\frac{x}{|x|}, & \text { if } x<0 \\ -1, & \text { if } x \geq 0\end{cases}$

$f(x)= \begin{cases}x+1, & \text { if } x \geq 1 \\ x^2+1, \text { if } x<1\end{cases}$

$f(x)= \begin{cases}x^3-3, & \text { if } x \leq 2 \\ x^2+1, & \text { if } x>2\end{cases}$

$f(x)= \begin{cases}x^{10}-1, & \text { if } x \leq 1 \\ x^2, & \text { if } x>1\end{cases}$

Is the function defined by

$$
f(x)= \begin{cases}x+5, & \text { if } x \leq 1 \\ x-5, & \text { if } x>1\end{cases}
$$

a continuous function?

Discuss the continuity of the function $f$, where $f$ is defined by
$f(x)=\left\{\begin{array}{l}3, \text { if } 0 \leq x \leq 1 \\ 4, \text { if } 1<x<3 \\ 5, \text { if } 3 \leq x \leq 10\end{array}\right.$

Discuss the continuity of the function $f$, where $f$ is defined by
$f(x)= \begin{cases}2 x, & \text { if } x<0 \\ 0, & \text { if } 0 \leq x \leq 1 \\ 4 x, & \text { if } x>1\end{cases}$

Discuss the continuity of the function $f$, where $f$ is defined by
$f(x)= \begin{cases}-2, & \text { if } x \leq-1 \\ 2 x, & \text { if }-1<x \leq 1 \\ 2, & \text { if } x>1\end{cases}$

Discuss the continuity of the function $f$, where $f$ is defined by
Find the relationship between $a$ and $b$ so that the function $f$ defined by

$$
f(x)= \begin{cases}a x+1, & \text { if } x \leq 3 \\ b x+3, & \text { if } x>3\end{cases}
$$

is continuous at $x=3$.

Discuss the continuity of the function $f$, where $f$ is defined by
For what value of $\lambda$ is the function defined by

$$
f(x)= \begin{cases}\lambda\left(x^2-2 x\right), & \text { if } x \leq 0 \\ 4 x+1, & \text { if } x>0\end{cases}
$$

continuous at $x=0$ ? What about continuity at $x=1$ ?

Discuss the continuity of the function $f$, where $f$ is defined by
Show that the function defined by $g(x)=x-[x]$ is discontinuous at all integral points. Here $[x]$ denotes the greatest integer less than or equal to $x$.
\item[20.] Is the function defined by $f(x)=x^2-\sin x+5$ continuous at $x=\pi$ ?
\item[21.] Discuss the continuity of the following functions:
(a) $f(x)=\sin x+\cos x$
(b) $f(x)=\sin x-\cos x$
(c) $f(x)=\sin x \cdot \cos x$
\item[22.] Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
\item[23.] Find all points of discontinuity of $f$, where

$$
f(x)=\left\{\begin{array}{cl}
\frac{\sin x}{x}, & \text { if } x<0 \\
x+1, & \text { if } x \geq 0
\end{array}\right.
$$

\item[24.] Determine if $f$ defined by

$$
f(x)= \begin{cases}x^2 \sin \frac{1}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{cases}
$$

is a continuous function?

Discuss the continuity of the function $f$, where $f$ is defined by
Is the function defined by $f(x)=x^2-\sin x+5$ continuous at $x=\pi$ ?
\item[21.] Discuss the continuity of the following functions:
(a) $f(x)=\sin x+\cos x$
(b) $f(x)=\sin x-\cos x$
(c) $f(x)=\sin x \cdot \cos x$
\item[22.] Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
\item[23.] Find all points of discontinuity of $f$, where

$$
f(x)=\left\{\begin{array}{cl}
\frac{\sin x}{x}, & \text { if } x<0 \\
x+1, & \text { if } x \geq 0
\end{array}\right.
$$

\item[24.] Determine if $f$ defined by

$$
f(x)= \begin{cases}x^2 \sin \frac{1}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{cases}
$$

is a continuous function?

Discuss the continuity of the following functions:
(a) $f(x)=\sin x+\cos x$
(b) $f(x)=\sin x-\cos x$
(c) $f(x)=\sin x \cdot \cos x$

Discuss the continuity of the cosine, cosecant, secant and
cotangent functions.

Discuss the continuity of the cosine, cosecant, secant and
cotangent functions.

Discuss the continuity of the cosine, cosecant, secant and
cotangent functions.

Discuss the continuity of the cosine, cosecant, secant and
cotangent functions.

Find all points of discontinuity of $f$, where

$$
f(x)=\left\{\begin{array}{cl}
\frac{\sin x}{x}, & \text { if } x<0 \\
x+1, & \text { if } x \geq 0
\end{array}\right.
$$

Determine if $f$ defined by

$$
f(x)= \begin{cases}x^2 \sin \frac{1}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{cases}
$$

is a continuous function?

Examine the continuity of $f$, where $f$ is defined by

$$
f(x)= \begin{cases}\sin x-\cos x, & \text { if } x \neq 0 \\ -1, & \text { if } x=0\end{cases}
$$

Find the values of $k$ so that the function $f$ is continuous at the indicated point
$f(x)=\left\{\begin{array}{ll}\frac{k \cos x}{\pi-2 x}, & \text { if } x \neq \frac{\pi}{2} \\ 3, & \text { if } x=\frac{\pi}{2}\end{array} \quad\right.$ at $x=\frac{\pi}{2}$

Find the values of $k$ so that the function $f$ is continuous at the indicated point
$f(x)=\left\{\begin{array}{ll}k x^2, & \text { if } x \leq 2 \\ 3, & \text { if } x>2\end{array} \quad\right.$ at $x=2$

Find the values of $k$ so that the function $f$ is continuous at the indicated point
$f(x)=\left\{\begin{array}{ll}k x+1, & \text { if } x \leq \pi \\ \cos x, & \text { if } x>\pi\end{array} \quad\right.$ at $x=\pi$

Find the values of $k$ so that the function $f$ is continuous at the indicated point
$f(x)=\left\{\begin{array}{ll}k x+1, & \text { if } x \leq 5 \\ 3 x-5, & \text { if } x>5\end{array} \quad\right.$ at $x=5$

Find the values of $a$ and $b$ such that the function defined by

$$
f(x)= \begin{cases}5, & \text { if } x \leq 2 \\ a x+b, & \text { if } 2<x<10 \\ 21, & \text { if } x \geq 10\end{cases}
$$

is a continuous function.

Show that the function defined by $f(x)=\cos \left(x^2\right)$ is a continuous function.

Show that the function defined by $f(x)=|\cos x|$ is a continuous function.

Examine that $\sin |x|$ is a continuous function.

Find all the points of discontinuity of $f$ defined by $f(x)=|x|-|x+1|$.

Differentiate the functions with respect to $x$
$\sin \left(x^2+5\right)$

Differentiate the functions with respect to $x$
$\cos (\sin x)$

Differentiate the functions with respect to $x$
$\sin (a x+b)$

Differentiate the functions with respect to $x$
$\sec (\tan (\sqrt{x}))$

Differentiate the functions with respect to $x$
$\frac{\sin (a x+b)}{\cos (c x+d)}$

Differentiate the functions with respect to $x$
$\cos x^3 \cdot \sin ^2\left(x^5\right)$

Differentiate the functions with respect to $x$
$2 \sqrt{\cot \left(x^2\right)}$

Differentiate the functions with respect to $x$
$\cos (\sqrt{x})$

Prove that the function $f$ given by

$$
f(x)=|x-1|, x \in \mathbf{R}
$$

is not differentiable at $x=1$.

Prove that the greatest integer function defined by

$$
f(x)=[x], 0<x<3
$$

is not differentiable at $x=1$ and $x=2$.

Find $\frac{d y}{d x}$ in the following:
$2 x+3 y=\sin x$

Find $\frac{d y}{d x}$ in the following:
$2 x+3 y=\sin y$
3. $a x+b y^2=\cos y$
4. $x y+y^2=\tan x+y$
5. $x^2+x y+y^2=100$
6. $x^3+x^2 y+x y^2+y^3=81$
7. $\sin ^2 y+\cos x y=\kappa$
8. $\sin ^2 x+\cos ^2 y=1$
9. $y=\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$
10. $y=\tan ^{-1}\left(\frac{3 x-x^3}{1-3 x^2}\right),-\frac{1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}$
11. $y=\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0<x<1$
12. $y=\sin ^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0<x<1$
13. $y=\cos ^{-1}\left(\frac{2 x}{1+x^2}\right),-1<x<1$
14. $y=\sin ^{-1}\left(2 x \sqrt{1-x^2}\right),-\frac{1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}$
15. $y=\sec ^{-1}\left(\frac{1}{2 x^2-1}\right), 0<x<\frac{1}{\sqrt{2}}$

Find $\frac{d y}{d x}$ in the following:
$a x+b y^2=\cos y$
4. $x y+y^2=\tan x+y$
5. $x^2+x y+y^2=100$
6. $x^3+x^2 y+x y^2+y^3=81$
7. $\sin ^2 y+\cos x y=\kappa$
8. $\sin ^2 x+\cos ^2 y=1$
9. $y=\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$
10. $y=\tan ^{-1}\left(\frac{3 x-x^3}{1-3 x^2}\right),-\frac{1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}$
11. $y=\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0<x<1$
12. $y=\sin ^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0<x<1$
13. $y=\cos ^{-1}\left(\frac{2 x}{1+x^2}\right),-1<x<1$
14. $y=\sin ^{-1}\left(2 x \sqrt{1-x^2}\right),-\frac{1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}$
15. $y=\sec ^{-1}\left(\frac{1}{2 x^2-1}\right), 0<x<\frac{1}{\sqrt{2}}$

Find $\frac{d y}{d x}$ in the following:
$x y+y^2=\tan x+y$

Find $\frac{d y}{d x}$ in the following:
$x^2+x y+y^2=100$

Find $\frac{d y}{d x}$ in the following:
$x^3+x^2 y+x y^2+y^3=81$

Find $\frac{d y}{d x}$ in the following:
$\sin ^2 y+\cos x y=\kappa$

Find $\frac{d y}{d x}$ in the following:
$\sin ^2 x+\cos ^2 y=1$

Find $\frac{d y}{d x}$ in the following:
$y=\sin ^{-1}\left(\frac{2 x}{1+x^2}\right)$

Find $\frac{d y}{d x}$ in the following:
$y=\tan ^{-1}\left(\frac{3 x-x^3}{1-3 x^2}\right),-\frac{1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}$

Find $\frac{d y}{d x}$ in the following:
$y=\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0<x<1$

Find $\frac{d y}{d x}$ in the following:
$y=\sin ^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0<x<1$

Find $\frac{d y}{d x}$ in the following:
$y=\cos ^{-1}\left(\frac{2 x}{1+x^2}\right),-1<x<1$

Find $\frac{d y}{d x}$ in the following:
$y=\sin ^{-1}\left(2 x \sqrt{1-x^2}\right),-\frac{1}{\sqrt{2}}<x<\frac{1}{\sqrt{2}}$

Find $\frac{d y}{d x}$ in the following:
$y=\sec ^{-1}\left(\frac{1}{2 x^2-1}\right), 0<x<\frac{1}{\sqrt{2}}$

Differentiate the following w.r.t. $x$ :
$\frac{e^x}{\sin x}$

Differentiate the following w.r.t. $x$ :
$e^{\sin ^{-1} x}$

Differentiate the following w.r.t. $x$ :
$e^{x^3}$

Differentiate the following w.r.t. $x$ :
$\sin \left(\tan ^{-1} e^{-x}\right)$

Differentiate the following w.r.t. $x$ :
$\log \left(\cos e^x\right)$

Differentiate the following w.r.t. $x$ :
$e^x+e^{x^2}+\ldots+e^{x^5}$

Differentiate the following w.r.t. $x$ :
$\sqrt{e^{\sqrt{x}}}, x>0$

Differentiate the following w.r.t. $x$ :
$\log (\log x), x>1$

Differentiate the following w.r.t. $x$ :
$\frac{\cos x}{\log x}, x>0$

Differentiate the following w.r.t. $x$ :
$\cos \left(\log x+e^x\right), x>0$

Ex 5.5, 1 teackoo
Differentiate the functions in, cos x .cos 2x .cos 3x
Let
y= cosx.cos 2x .cos 3x
Taking log both sides
log y = log (cos x. cos 2x . cos 3x)
log y = log (cos x) + log (cos 2x) + log (cos 3x)
Differentiating both sides w.7r. t. x.
dllogy) d(log(cos x) + log (cos 2x) + log (cos 3x))
ax 7 dx
d(log y) (2) _ d(log (cos x)) + d(log (cos 2x)) + d(log (cos 3x)
dx dy. ~ dx dx dx

Ex 5.5, 2 teachoo.com
. . - («-1)@ - 2)
Differentiate the functions in, @-De-D@—5)
= @-1)@-2)
let y= Nee -N@—5)
i
_ (x - 1) - 2) 2
y= (q ~2)a-)a- 5)
Taking log both sides
i
_ @-1)@- 2) 2
logy = log (q “DE -De 5)
1 @-1)@—-2) b
== |} As l = bl
logy 5 log (Sie oes) (As log(a”) b log a)

Ex 5.5, 3 teachoo.com
Differentiate the functions in, (log x)°°S*
Let y = (logx)°°S*
Taking log both sides
logy =log (logx)*°**
logy = cos x .log (log x) (As log(a’) = bloga)
Differentiating both sides w.7r. t. x.
d(logy) _ d(cosx.log (log x))
dx ~ dx
d(log y) (2) _ a(cosx . log (log x))
dx dy. ~ dx
d(log y) (2) _ d(cos x. log (log x))
dx dx} dx

teachoo.com

Ex 5.5, 4
Differentiate the functionsin, x* - 25™*
Let y=x* - 28inx
Let u=x* ,v = 25In*

yru-v
Differentiating both sides w. r. t. x.

dy d(u-v)

dx dx

dy _ du dv

dx dx dx

teachoo.com

Ex5.5,5
Differentiate the functions in, (x + 3)*.(x + 4)?.(x« + 5)4
Let y=(x + 3)*.(x + 4)°.(x + 5)4
Taking log both sides
logy= log (( + 3)?.( + 4)3.(@ + 5)4)
logy = log (x + 3)° +log (x + 4)? +log (x + 5)*
logy= 2log (x + 3) +3log(x + 4)+4log (x + 5)
Differentiating both sides w.r.t. x.
dQogy) _ d(2log(v+3) + 3log (x +4) + 4 log (x +5))

dx ~ dx

Ex 5.5, 6 teachoo.com
1\* i
Differentiate the functions in, (x + =) + x(t + 2)
x 1
Let y= (x+=) + x(*3)
x 1
Let u =(x+=) vexltts)
x.
y=utv
Differentiating both sides w.r. t. x.
dy d(tu+v)
dx dx
dy _ du + dv
dx dx dx

Ex5.5,7 teachoo.com
Differentiate the functions in, (logx)* + x!0&*
Let y = (log x)*+ x!08*
Let u = (logx)*, v= x!08*
y=utv
Differentiating both sides w.r. t. x.
dy d(u+v)
dx — dx
dy _ du + dv
dx dx dx

EX5.5, 8 teachoo.com
Differentiate the functions in, (sinx)*+ sin-t x
Let y = (sinx)* + sin-1-¥x
Letu =(sinx)* & v = sin-t-¥x
y=utov
Differentiating both sides w.r. t. x.
dy _d(utv)
dx — dx
dy _ du + dv
dx dx dx

teachoo.com

Ex 5.5,9
Differentiate the functions in, x°!"*+ (sin x)°S*
Let y= x5 * + (sin x)oS*
Letu = x5™* & vy = (sin x)°°S*
“y=utv
Differentiating both sides w.r. t. x.

dy _d(utv)

dx dx

dy du, dv

dx dx dx

teachoo.com
Ex 5.5, 10
. . . . +1
Differentiate the functions in, x* °8* + roar
eat Important-
Let y=x% 605% 4 ——
x- 1
xX COSX +1
letu =x &v ==
x1
“yu +v
Differentiating both sides w.r. t. x.
dy _d(utv)
dx dx
dy _ du + dv
dx dx dx

Ex 5.5, 11 teachoo.com
a
Differentiate the functions in, (x cosx)* +(x sinx) x
1
y=(xcosx)* + (sinx) =
1
Let u =(xcosx)*,v=(xsinx)*
yeoutyv
Differentiating both sides w. 7. t. x.
dy _d (u+v)
dx dx
dy _ du + dv
dx dx dx

Ex 5.5, 12 teachoo.com
+ dy + + x
Find x of the functionsin, x” +y*=1
xv ey*al
Let u =x” ,v=y*
Hence,
utv=l1
Differentiating both sides w.r. t.x.
dwvtu)_ dQ)
dx ~ dx
dv du wae .
—+—-=0 (Derivative of constant is 0}
dx dx

teachoo.com

Ex 5.5, 13
. ay oo x
Find a of the functionsin, y* =x”
Given,

yr ax?
Taking log both sides

log (y*) = log (x”)

x .log y = y.logx (As log(a”) = b.log a)
Differentiating both sides w.r. t. x.

d(x.logy) _ d(y.log x)
dx ~ dx

teachoo.com
Ex 5.5, 14
Find “ of the functions in, (cos x )” = (cos y )*
Given
(cos x)” = (cos y)*
Taking log both sides
log (cos x)” = log (cos y)*
y .log (cos x) = x. log(cos y) (As log(a”) = b.loga)
Differentiating both sides w. r. t. x.
d(y.log(cosx)) _ d(x. log(cos y))
dx ~ dx

teackoo.com
Ex 5.5, 15
Find ie of the functions in, xy = e@-¥)
Given
xy = e&-»)

Taking log both sides

log (xy) = loge@-»)

log (xy) =(x — y) loge (As log(a?) = b.log a)

logx + logy =(x —y) (1) (As loge = 1)

logx + logy =(x —y)

Ex 5.5, 16 teachoo.com
Find the derivative of the function given by f (x) = (1 +x) (1+ x7)
(1+x*) (1+ x8) and hence find f’(1).
Given
f@)=(4xn04+2x90 +2490 + x9)
Lety = (1+x)(14+x7)(1+ x71 + x8)
Taking log both sides
logy = log(l+x)+x7)1 +241 + x9)
(As log(ab) = log at log b)
logy = log (1 + x) + log(1 + x”) + log(1 + x*) + log (1 + x°)

Ex5.5, 17 teachoo.com
Differentiate (x? — 5x + 8) (x3 + 7x +9)
(ii) by expanding the product to obtain a single polynomial.
By Expanding the product to obtain a single polynomial .
y = (x? -5x +8) (x3 + 7x49)
y =x? (x3 + 7x4 9) —Sx(x3 + 7x49) 48 (x2 + 7x49)
y=x>+ 7x3 + 9x? — 5x4 — 35x2 — 45x 4 8x3 + 56x +472
y =x>— 5x44 15x39 — 26x27 4+ 11x + 72
Differentiating both sides w.r. t. x.
dy — d(x? — 5x44 15x3— 26x? + 11x +72)
dx dx
dy d(x) _ a(ex*) + d(15x7) _ d(26x?) + d(11x) + d(72)
dx dx dx dx dx dx dx

Ex5.5, 18 teachoo.com
If u, v and w are functions of x, then show that
d du dv dw
m (u. v.w)= vewtu. - wtu.v i
in two ways - first by repeated application of product rule,
second by logarithmic differentiation.
By product Rule
Let y = uvw
Differentiating both sides w.r. t.x.
dy _ d(uvw)
dx dx
dy _ d((uv) w)
dx dx

Ex 5.6, 1 teackoo
If x and y are connected parametrically by the equations
without eliminating the parameter, Find 2, x=2at?, y=at*
Here
ay
dy _ dt
dx ax
dt
_ dy _ dx
Calculating dt Calculating at
y =at* x =2at?
dy _ 4-1 dx _
dt = 4Aat dt =2 X 2at
= 4at3 = 4at

Ex 5.6, 2 teachoo.com
If x and y are connected parametrically by the equations without
eliminating the parameter, Find 2, x=acos@,y=bcosé
Here
dy
dy _ 40
dx ax
a0
Calculating = Calculating
do alcu ating 75
y =bcos@ x= acos@
dy _ a(bcos 0) dx _ d(acos 8)
do do do do
dy _ —e«j dx _ _
ae 7 Oh sin@) a 7a sin9)
dy _ : dx _ .
aa = bsin@ qa 7 asind

Ex 5.6, 3 teachoo.com
If x and y are connected parametrically by the equations without
eliminating the parameter, Find * x =sint,y = cos2t
Here,
dy
dy _ ae
ax
dt
. dy . dx
Calculating a Calculating a
dy _ d(cos 2t) dx] d(sin t)
dt” ~— at dt — dt
dy __ ax _
a sin 2t .2 ae 7 cost
d. :
“ =-2 sin2t
dt

Ex 5.6, 4 teachoo.com
If x and y are connected parametrically by the equations
without eliminating the parameter, Find, x= 4t,y = :
Here
dy
ay _ at
dx &
dt
ay 4
Calculating dt Calculating =
dy = <() dx_ d(at)
dt dt \t a = a
dy d {1
““=4—f- dx
dt 4k (7) —=4
dt
ay 4
att

Ex5.6,5 teachoo.com
If x and y are connected parametrically by the equations
_ 4d
without eliminating the parameter, Find ae
x = cos@- cos26,y = sin@- sin2@
Here
dy
dy _ ao
dx ak
ao
~ dy ax
Calculating = Calculating 7,
ay _ d(sin 6 - sin 26) dx _ d(cos@~cos 20)
do ae do” ao
ay = d(sin 0) _ d(sin 26) dx _ d(cos@) _ d{cos 26)
dé dé dé aos do
dy d
49 = COSA — cos26.2 =~ sind — (sin 20 .2
ay _ 0-2 20 dx . .
ae £08 cos ao = 7 Sin@ + 2sin26

Ex 5.6, 6 teachoo.com
If x and y are connected parametrically by the equations
without eliminating the parameter, Find 2,
x = a(@-sin@),y = a(1 + cos@)
Here
dy
dy _ do
dx ax
doe

Ex 5.6, 7 teachoo.com
If x and y are connected parametrically by the equations
without eliminating the parameter, Find *
_ _sinst _ cost
x= vcos 2t” y= ¥cos 2
Here,
ay
ay _ ae
ax ax
at

teachoo.com
Ex 5.6, 8
If x and y are connected parametrically by the equations
without eliminating the parameter, Find ae
x= a(cost + log tan=), =asint
Here
dy
ay _ de
dx a
dt

Ex 5.6, 9 teachoo.com
If x and y are connected parametrically by the equations
without eliminating the parameter, Find,
x=a sec 6,y = btand
Here
dy
dy _ dé
dx ax
dé
. ay . dx
Calculating 70 Calculating qo
dy _ d(btan 6) dx _ d(asec 8)
de°—sé«O ao a0
ay _ b d(tan @) ax _ a(sec 0)
ado” a0 ao" a6
ay _ 2 ax _
ap = D-Sec 0 ap 7 2 ([email protected]®)

teachoo.com
Ex 5.6, 10
If x and y are connected parametrically by the equations
without eliminating the parameter, Find a
x = a(cos@ + @sin@),y = a (sind - 6cos@)
Here
dy
ay _ ae
dx ax
ao

€x 5.6, 11 teachoo.com
-6,
—— = a
fx =yasr tt, y=vyaes"t, show that > =-2
Here
dy
ay _ ae
dx ax
dt

teachoo

Ex 5.7,1
Find the second order derivatives of the function x? + 3x +2
Let y =x? 43x+2
Differentiating w.r.t.x

dy _ d(x? +3x+2)

dx dx

2

dx ax ax ax

d

i 2x4+340

dx

dy

a 2x +3

teachoo.com

Ex 5.7, 2
Find the second order derivatives of the function x2°
Let y = x2°
Differentiating w.r.t.x

dy _ d(x?°)

dx dx

PY = 29x20-1

dx

d

> = 20x"

dx
Again Differentiating w.r.t.x

d (2) _ a (20x1)

dx \ax} ~ ax

teachoo.com

Ex 5.7, 3
Find the second order derivatives of the function x. cos x
Let y =x. cos x
Differentiating w.r.t.x .

dy _ d(x. cos x)

dx dx
Using Product Rule
As (uv)’= uv + vu

dy _ d(x) d(cos x)

de dy (OSH +

dy _ _

ae = COS% +(-—sinx).x

Ex5.7, 4 teachoo.com
Find the second order derivatives of the function log x
Let y = log x
Differentiating w.r.t.x .

dy _ d(log x)

dx ~ dx

dy oi

dx x
Again Differentiating w.r.t.x

ila) =a)

dx \dx} ~ dx\x

teachoo.com

Ex 5.7,5
Find the second order derivatives of the function x° log x
Let y=x? logx
Differentiating w.r.t.x .

dy _ d(x* logx)

dx dx
using product rule in x?
log x.
As (uv)’= wv + vu
where u =x? & v=log x

dy — a(x?) d(log x)

==.) +

dx dx 98% dx *

Ex 5.7, 6 teachoo.com
Find the second order derivatives of the function e* sin5x
Let y= e* sin 5x
Differentiating w.r.t.x .

dy _ d(e* sin 5x)

dx” dx
using product rule in e* sin 5x
As (uv)’= uv + vu

d d(e~ d(sin5 x

o. ae") .sin 5x+ asin 5%) ax

dx dx dx

d d(5x

& =e* .sin 5x + cos 5x 46%) e*

dx dx

d

& = e*. sin 5x+5.e%. cos 5x

dx

EX5.7,7 teachoo.com
Find the second order derivatives of the function e® cos 3x
Lety = e* cos3x
Differentiating w.r.t.x.
dy _ de® cos 3x)
dx dx
Using product rule in e® cos 3x.
As (uv)’= wv + vu
where u = e* & v=cos 3x
dy _ a(e®) d(cos3x)
— = —~— .cos 3x + ———— .e™*
dx dx dx
dy 6x 46%) . d(x) 6x
—= _—. +(- —_.
x 78 am oS 3x + (—sin 3x) me

Ex 5.7,8 teachoo.com
Find the second order derivatives of the function tan7+ x
Let y= tan-tx
Differentiating w.r.t.x .

dy _ d(tan7+ x)

dx dx

dy _ 1

dx 1+x2
Again Differentiating w.r.t.x

i (as) > a (aa)

dx\dx/ dx \1+x?

dy d ( 1 )

dx? ~ dx \1+x2

Ex5.7,9 teachoo.com
Find the second order derivatives of the function log (log x)
Let y=log (log x)
Differentiating w.r.t.x .

dy _ d(log dog x))

dx — dx

dy 1 d(log x)

dx logx’ dx

dy _ 1 1

dx logx ‘x

dy _ 1

dx — x.logx

teachoo.com

Ex 5.7, 10
Find the second order derivatives of the function sin (log x)
Lety= sin (log x)
Differentiating w.r.t.x .

dy _ d( sin (log x))

dx — dx

dy _ d(log x)

a 7 cos(log x) a

dy _ 1

ms cos(logx) 5

dy _ cos(log x)

dx x

Ex 5.7, 11 teachoo.com
. ay
Ify =5 cosx —3sinx ,prove that me te 0
y=5 cosx —3sinx
Differentiating w.r.t.x
dy _ a(S cos x—3 sin x)
dx dx
dy _d(Scosx) dQ@sin~x)
dx dx dx
d
& = -Ssinx -3cosx
dx
Again Differentiating w.r.t.x
d (2) _ d(—5sin x — 3cos x)
dx\dx} ~ dx

teachoo.com
Ex 5.7, 12
-1 . ey,

If y=cos~* x, Find qa interms of y alone.
Let y=cos-tx
Differentiating w.r.t.x

dy _ d(cos7* x)

dx — ax

dy = =1 A a(cos-tx) — =1

dx V1—x2 rn
Again Differentiating w.r.t.x

a (2) 4 (=)

dx\dx} ~ dx\V1—x2

Ex5.7, 13 teachoo.com
If y=3 cos (logx) + 4 sin (logx),
show that x?y, +xy,+y=0
y =3cos (logx) + 4 sin (log x)
Differentiating w.r.t.x
dy _ 3 1 1
an =-3 sin (log x) x= +4 cos (log x} x=
oy 3 sin (log x} + 4 cos (log x}
X 3 = 73 sin (log x cos (log
Differentiating w.r.t.x
dy)" , ,
(x ~) = (-3 sin (log x))’ + (4 cos (log x})

Ex 5.7, 14 teachoo.com
TF,
a a.
If y =Ae™* + Be™, show that -(m+n) a +mny =0
y = Ae™* +Be™
Differentiating w.r.t.x
dy _ a(Ae™ + Be?)
dx dx
dy _ ad(Ae™*) + d(Be™™)
dx dx dx
dy =A.em d(mx) +B.e™ d(nx)
dx . “dx . dx
d
= Ale™,) m+B.e™ n
dx
d
<= Ame™ +Bne™
dx

Ex 5.7, 15 teachoo.com
If y = 500e7*+ 600e~”*, show that < = 49
y = 500e7*+ 600e-”*
Differentiating w.r.t.x
dx” dx
dy _ d(500e7*) + d (600e~7*)
ax ~ dx ax
dy _ d(e”*) d(e-7*)
7, = 500——— _ + 600 ——
dy _ x A(7%) ay 4 (-72)
Ig 7 300.€%. = + 600.e°7* . —

Ex 5.7, 16 (Method 1) teachoo.com
y - &y _ (ay?
Ife” (x+ 1) = 1, showthat = = (2)
We need to show that
ay (2y
dx? ~~ \ax
ex+t)H=1
Differentiating w.r.t.x
a(e¥(x+1)) _ aq)
dx "ax
a(eY (x+1)) _ 0
dx ~

teachoo.com

Ex 5.7, 17 (Method 1)
If y = (tan-1 x)”, show that (x? + 1)? y,+ 2x (x? +1) y,=2
We have

y = (tan? x)?
Differentiating w.r.t.x

"=2tantxx a

ye 1+ x4

(14+x7)y’ =2tantx
Again differentiating w. r. t.x

[ (1+ x*)I =2x 5

y 1+x? 1+x?

b+ =

y 14+ x?

Check the continuity of the function $f$ given by $f(x)=2 x+3$ at $x=1$.

Examine whether the function $f$ given by $f(x)=x^2$ is continuous at $x=0$.

Discuss the continuity of the function $f$ given by $f(x)=|x|$ at $x=0$.

Example 4 Show that the function $f$ given by

$$
f(x)= \begin{cases}x^3+3, & \text { if } x \neq 0 \\ 1, & \text { if } x=0\end{cases}
$$

is not continuous at $x=0$.

Check the points where the constant function $f(x)=k$ is continuous.

Prove that the identity function on real numbers given by $f(x)=x$ is continuous at every real number.

Is the function defined by $f(x)=|x|$, a continuous function?

Discuss the continuity of the function $f$ given by $f(x)=x^3+x^2-1$.

Discuss the continuity of the function $f$ defined by $f(x)=\frac{1}{x}, x \neq 0$.

Discuss the continuity of the function $f$ defined by

$$
f(x)=\left\{\begin{array}{l}
x+2, \text { if } x \leq 1 \\
x-2, \text { if } x>1
\end{array}\right.
$$

Find all the points of discontinuity of the function $f$ defined by

$$
f(x)=\left\{\begin{array}{cc}
x+2, & \text { if } x<1 \\
0, & \text { if } x=1 \\
x-2, & \text { if } x>1
\end{array}\right.
$$

Discuss the continuity of the function defined by

$$
f(x)=\left\{\begin{array}{r}
x+2, \text { if } x<0 \\
-x+2, \text { if } x>0
\end{array}\right.
$$

Discuss the continuity of the function $f$ given by

$$
f(x)= \begin{cases}x, & \text { if } x \geq 0 \tag{-2,4}\\ x^2, & \text { if } x<0\end{cases}
$$

Show that every polynomial function is continuous.

Find all the points of discontinuity of the greatest integer function defined by $f(x)=[x]$, where $[x]$ denotes the greatest integer less than or equal to $x$.

Prove that every rational function is continuous.

Discuss the continuity of sine function.

Prove that the function defined by $f(x)=\tan x$ is a continuous function.

Show that the function defined by $f(x)=\sin \left(x^2\right)$ is a continuous function.

Show that the function $f$ defined by

$$
f(x)=|1-x+|x||,
$$

where $x$ is any real number, is a continuous function.

teachoo.com

Example 21
Find the derivative of the function given by f (x) = sin(x?).
Let y= sin(x?)
We need to find derivative of y, w.r.t.x
_. dy — d(sin x?)
re. dx dx

= cosy? , 22

= COSX? .

= cosx? . (2x?71)

=cosx? (2x)

=2x.cos x?

teachoo.com
Example 22
ay. _
Find ax ifx — yen.
(x-y)=1
Differentiating both sides w.rt x
d(x-y)_ dn
dx ~ dx
dx dy _ .
i an 0 (As 7 is constant)
d
1-2=0
dx
dy
dx 1

Example 23 teachoo.com
Find = , if y+siny=cosx
aitytsiny=
ytsiny =cos x
Differentiating both sides by x
dy + a(siny) _ d(cos x)
dx dx —s dx
d d(sin .
ay , Hsin ¥) _ _ gin y
dx dx
d d(sin d .
a+ asin y) 2 =-sinx
dx dy dx
d d .
+ cosy — =-sinx
dx dx

Example 24 teackoo.com
Find the derivative of f given by f (x) = sin-' x assuming it exists.
f@)=sin 1x
lety = sin"1x
siny=x
x=siny
Differentiating both sides w.r.t.x
dx_d (sin y)
dx dx
d({siny) dy
t= OO xe
dx dy

teachoo.co
Example 25 "
: I

Is it true thatx =e °” forall real x?

x=e log x
Forx=0

O= elog 0
But log 0 is not defined.
Hence, the equation is not defined for x = 0

Forx <0
x=e log x

But log x is not defined for negative numbers
Hence, equation is not defined for x <0

teachoo.com

Example 26
Differentiate the following w.rt. x:
{i)e*
Llety =e*
Differentiating both sides w. r. t.x

dy _ d(e*)

dx dx

dy =e d(—x)

dx “dx

dy x

a 7 e (-1).

dy -x

dx e

Example 26 Differentiate the following w.r.t. x: (ii) sin⁡(log⁡𝑥), 𝑥 > 0
Let 𝑦 =sin⁡(log⁡𝑥)

Example 26 Differentiate the following w.r.t. x: (iii) 〖𝑐𝑜𝑠〗^(−1) "(ex)" Let 𝑦 = 〖𝑐𝑜𝑠〗^(−1) "(ex)"

Example 26 Differentiate the following w.r.t. x: (iv) 𝑒𝑐𝑜𝑠 𝑥Let 𝑦 = 𝑒^cos⁡𝑥

Example 27 teachoo.com
_ 2
Differentiate jee w.r.t.x.
3x*+4x4+5
(x — 3) (x2 +4)
Let y= =e
3Bx°+4x4+5
Taking log on both sides
_ [a-G*+4)
logy =log 3x74 4x45
2 1
_ (x — 3) (x? + 2)
logy = log ( Bx7+4x4+5
aNog SIDE Using loga® = bloga
logy =7108 Garazas ‘(Using log Ba)

Example 28 teachoo.com
Differentiate a* w.r.t.x, where a is a positive constant.
Let y =a*
Taking log on both sides

logy = loga*

logy =xloga (log a? = bloga)
Differentiating both sides w.r.t.x

ddogy) _4

a ae (xlog a)

ddogy) _ ax

a. loga (=)

ddogy) _ loga

dx ='0g

Example 29 teachoo.com
Differentiate xS™*, x > O w.r.tx.
Let y= x5in*
Taking log both sides
logy = log x5in *
logy=sinx.logx (loga® =bloga)
Differentiating w.r.t.x
d(logy)_d |.
ax dx (sin x log x)
By product Rule
(uv)’=u’v + vu
where u = sin x & v= log x

teachoo.
Example 30 CACHOO.LOM
Find &, if y* 42% +x% =a?
dx’ .
Let u=y*, vex” &we= x*
Now,
ut+v+we=a
Differentiating w.r.t.x
d(u+viw) _ aca?)
dx "dx
au), a) | dw) _ (As a” is constant) (1)
dx dx dx
We will calculate derivative of u, v & w separately .

Example 31 teachoo.com
. dy . .
Find a if x = acos6,y = asin®.
dx
Here
dy
dy _ a6
dx ax
ao
: dy . dx
Calculating 70 Calculating aa
y =asin®8 y =acosd
dy _ d(asin @) dy _ d(acos 6)
do” ~— do” do
d d .
“= acos@ “= -asin@
do do

Example 32 teachoo.com
. dy. _ 2 _
Find —, ifx = at?,y = 2at.
dx
Here
dy
dy _ at
dx ax
dt
. ay . dx
Calculating at Calculating 77
y = 2at x =at?
dy _ d(2at) dx _ d(at’)
dt dt dt ~ dt
dy 94 a dx a)
ae 24a a ae
dy dx
at =2Za at =2at

Example 33 teachoo.com
Find =, if x = a(@+sin@),y = a(1- cos@)
Here
ay
ay _ a6
dx ax
ae
_ ay da
Calculating de Calculating a
y =a(1- cos@) x =a(O+sin8@)
dy _ a(a (1~ cos 6)) dx d(a(@+sin@))
d@ d@ ae = de
dy . .
— =a(0 —(-siné@) ax _ | (ao, alsin 8)
ae ( ) aa 72 (aot do )
dy _ . d
qq 7 @(sin8) 7a +0088)

Example 34 (Method 1) teachoo.com
2 2 2
Find =, if xe+y3 =a3,
dx
2 2 2
x3+ y3 = a3
Differentiating w.rt. x
2 2 2
d(x3) 4 d(y3) _ d(a3)
dx dx dx
2 2
20 =-141 d(y3) d
22-14 403) ay 9
3 dx dy
1 2
2 = dvy3 da
2,3, a0), ay _ 9
3 dy dx
2. 2.24. ay
—-x3 +-y3 x—=0
3 3 ¥ dx

Example 35 teachoo.com
Find vy if y = x3 +tanx
ind Saif y = .
y = x3+tanx
Differentiating w.r.t.x
dy _ d(x3+tanx)
dx ax
dy _ d(x) + d(tan x)
dx ax ax
d
23x? + sec?x
dx
Again Differentiating w.r.t.x
a?y _ da (3x? tsec? x)
ax2 — ax

Example 36 teachoo.com
. ay
lfy = Asinx + Bcosx, then prove that ae tYe 0.
y = Asinx +Bcosx
Differentiating w.r.t.x
dy _— d(Asinx+Bcosx)
dx — dx
dy _ d(Asin x) + d(B cos x)
dx dx dx
d. d(sin x d(cos x
dy _, dsinx) |, d(cosx)
dx dx dx
d
& =Acosx +B (- sin x}
dx
dy .
— =Acosx -Bsinx
dx

Example 37 teachoo.com
ay ay
= 2K 3K —- ~-5— =

If y = 3e* + 2e**, prove that a Sox + 6y =0.,
Given,

y = 3e% + 263
Differentiating w.r.t.x

dy _ d(e*+ 2e*)

dx dx

dy de”) sae)

dx dx dx

dy _ ox d(2x) yy AGBx)

am” 3.e a +2.e3

d

& = 3.6.24 2.¢3%.3

dx

Example 38 (Method 1) teachoo.com
—_ ay dy
= 1 _ y2y OY _y
If y= sin~*x, show that (1 - x7) a ax oO.
We have
y=sin tx
Differentiating w.r.t.x
dy _ a(sin™*x)
dx — dx
dy - 1 (A a{sin"tx) 4 )
dx V1—x2 8 ie ~ Vix?
V(1—2?) y'=4
Squaring both sides

Example 39 teachoo.com
Differentiate w.rt. x, the following function:
. 1
(i) V38x+2 + Ea
L
Let y= V3x + 2+
Differentiating w.r.t.x
a vax ++ ——)
dy _ vax? +4
dx dx
wa) . (Ee)
dy _ a(v¥3x+2) + 2x2 + 4
dx dx dx
a1
dy _ d(v3x+2) + d(2x? +4)2
dx dx dx

Example 39 Differentiate w.r.t. x, the following function: (ii) log7 (log x) y = log7 (log x)

Example 40 (Method 1) teachoocom
Differentiate the following w.r.t.x.
(i) cos~* (sin x)
Let f(x) = cos“ (sin x)
f (x) = cos™* (cos G - x)) (As sin@ =cos G - x))
f@)=5-*
Differentiating w.r.t.x
a G) d(x) (As 1 27 gn is constant)
, =—2/ _ oN
f ) ~ dx dx “ ,
P@)=0-1
f@=-t

Example 40 (Method 1) Differentiate the following w.r.t. x. (ii) tan −1 (sin⁡𝑥/( 1 +〖 cos〗⁡〖𝑥 〗 ))
Let 𝑓(𝑥) = tan −1 (𝒔𝒊𝒏⁡𝒙/( 1 +〖 𝒄𝒐𝒔〗⁡〖𝒙 〗 ))

Example 40 Differentiate the following w.r.t. x. (iii) sin^(−1) ((2^( 𝑥+1) )/( 1 +〖 4 〗^𝑥 ))
Let 𝑓(𝑥) = sin^(−1) ((2^( 𝑥+1) )/( 1 +〖 4 〗^𝑥 ))
𝑓(𝑥) = sin^(−1) ((2^( 𝑥). 2)/( 1 + (2^𝑥 )^2 ))

Example 41 teachoo.com
Find f‘(x) if f (x) = (sinx)S™* foralld<x<n.
Let y = (sinx)si"*
Taking log on both sides
log y = log (sin asin *)
log y = sinx . log (sin x) (As log(a”) = b. log a)
Differentiating both sides w. r. t. x
d(logy) _ a(sin x. log (sin x))
ax ~ ax
d(log y) (2) _ d(sin x. log (sin x))
dx dy ~ dx

Example 42 teachoo.com
sgt . dy
For a positive constant a find ae’ where
1 2
y =a'ti,and x = (t++)
Here
dy
dy _ ae
dx &
dt

teachoo.com
Example 43
Differentiate sin?x w.r.t.e°°S*:
Let u =sin’x & v =e°S*
We need to differentiate u wrt. v.
. du
ie, —
dv
Here,
du
du _ ik
dy ww
dx

Mise 1 teachoo.com
Differentiate w.rt. x the function,
(3x2 - 9x + 5)?
Let y = (3x2- 9x + 5)?
Differentiating w. r. £. x.
dy — a(3x?-9xt 5)?
dx dx
d(3x?-9x +5)
= 2_ 9-1
9x? - 9x + 5) . ax
d(3x? d(9. a(S
= 9(3x2 - 9x + 5° (2 — d(x) = )
dx dx dx
= 9(3x2- 9x + 5)®. (6x -9+0)
= 9(3x2- 9x + 5)®. (6x — 9)

Misc 2 teachoo.com
Differentiate w.r.t.x the function
sin? x + cos°x
Let y=sin? x + cos® x
Differentiating w. r. t. x.
a(( ingx) + 6
dy sin? x) + (cos® x }
dx dx
d( (sin? x d((cos® x
dx dx
d(sin x d(cos x.
= 3 sin’x asin) + 6 cosSx , ees ™)
dx dx
=3sin?x .cosx + 6 cos°x. (—sinx)

Misc 3 teachoo.com
Differentiate w.r.t.x the function, (5x)?°s 2*
Let y= (5x)300s 2x
Taking log on both sides
logy =log (5x)3°°s 2%
log y=3 cos 2x .log 5x (As log(a’) = bloga)
Differentiating both sides w.r. t. x
d(log y) _ dG cos 2x. log 5x)
dx ~ dx
d(log y) (2) _ a3 cos 2x. log 5x)
dx dy ~ dx

Misc 4 teachoo.com
Differentiate w. r.t. x the function,
sin (x ¥x),0 <x <1
Let y =sin™* (x yx)
1
y=sin' (x. x2)
— ein t ppt t%
y=sin™ (x "2)
3
y= sin} (x2)
Differentiating w. r. t. x
3
dy _ a{sin-2 (x2))
dx dx
d oe
y 1 d(x)2 4
“es , x (As d{sin x) 1 )
dx (2) ax dx Vi-x?
1-\x2.

. teachoo.co
Misc 5 "
Differenti he function, 2 ,- 2 2
ifferentiate w.7r.t.x the unction, 7 <x <
cos"
Let y = ——=
y V2X47
Differentiating both sides w.r. t. x
-1%
dy _ a [cos*>
dx dx V2X+7
Using Quotient rule
As (“) = ulv au
v v
where u = cos*> &vev2x+7

. teachoo.co
Mise 6 (Method 1) "
Differentiate w.rt. x the function,

pol vil+sinx +v1-sinx Q< <t
co ——_—__— ], xe

vit+sinx —v1—-sinx 2
Let y = cot7? Vv¥1l+sinx+V1- sinx
y= vitsinx—-y1-—sinx
Rationalizing the sum
fa (V1 + sin x + ¥1—sinx) y (iL + sinx + vi— sin x)
= co a ——
y (Vi+sinx-Vi-sinx) (Vi+tsinx + V1 -sinx)
2
t (V1 + sin x + V1 — sin x)

=co ee

y (VI + sinx - ¥1-sinx) (VI+sinx++¥1-sinx)

teachoo.com
Misc 7
Differentiate w.rt. x the function,
(log x) 8 >1
log x
Let y = (log x)
Taking log both sides
logy =log (Cog x) °8*)
log y =logx. log (log x) (As log(a”) = bloga)
Differentiating both sides w.r. t. x.
d(logy) _ dog x.log (log x))
ax ~ dx

Mise 8 teachoo.com

Differentiate w.7r.t.x the function,

cos (a cos x + b sin x), for some constant a and b.

Let y=cos (a cosx + bsinx)

Differentiating w.r.t.x.

dy _ 4(cos((acos x+b sin x))

ax dx

wy —sinx (a cosx + b sinx) _ Ma cosx+b sin)

dx dx
-_« : d(cos x) d(sin 2)
=—sinx(acosx+bsinx). (a. 2 +b a
=—sinx(acosx+bsinx). (a(- sinx) + b (cos x))

teachoo.co

Misc 9 "
Differentiate w.rt. x the function,
(sin x — cos x)6Sin ¥—C08 2) . <x< ae
Let y = (sinx — cos x)(sin *-c0s x)
Taking log on both sides

logy =log (sin x — cos x) in *—cos *)

log y = (sin x — cos x). log (sin x — cos x)
Differentiating both sides w.r. t. x.

d(logy) _ a((sin x — cos x). log(sin x — cos x))

dx ~ dx

Misc 10 teachoo.com
Differentiate w.rt. x the function,
x* + x* + a* + a*, forsome fixeda > Oandx> 0
Lety = x* + x7 + a*+ a?
Andletu = x*, v=x*, w=a*
Now,
y=ut+vtiwtast

Differentiating both sides w.r. t. x.

dy d(u+v+wta’)

dx dx

dy dtu) , dv) , dw) d(a*)

dx dx te bax + dx

Misc 11 teachoo.com
Differentiate w.rt. x the function,
x’-3 4 (¢ —3)" ,forx > 3
Let y =x7°-3 + (x — 3)?”
Andlet u=x* ~?,v=(x-3)*
Now,

y=ut+v
Differentiating both sides w.r. t. x.

dy _ d(tut+v)

dx dx

dy _ du + dv

dx dx dx

teachoo.com
Misc 12
Find 2 if y = 12 (1-cost),x=10 (t- sint),-—F <x< 5
Here,
dy
dy _ dt
dx dx
dt

Misc 13 teachoo.com
Find =, if y=sin-'x+sin'tV1—2x2,-1<x%<1
y=sinixtsintv1i-x*, -1<x<1
Puttingx = sin®@

y = sin-1 (sin@) + sin“! V1 — sin2@

y = 6+ sin"! Vcos26

y =@+sin™ (cos @)

y=@+sin* (sin G - 6)) (As cos @ = sin §-8))

we
y=9+(5-9)

Misc 14 teachoo.com
| Vf - _ dy -1
lfxJltytyvit+xe= 0, for 1<x<1, prove that — - Gay
xJjlt+ytyv1+x=0
x f/l+y =-yv1+x
Squaring both sides
2 2
(xf/1+y) = (-yv1+x)
2 2
x? (J/1+y ) = Gy? (WI+x)
2 = y2
x(1+y) = y* +x)
x? +xty = y? + yx

Misc 15 teachoo.com
If (x- a)? + (y- b)? = c2,forsomec > 0, prove that
3
dy 272
[+ @) ;
—qy ‘Isa constant independent of a and b.
x2
. . dy
First we will calculate —
dx
(x- a)? + (y- bY = ¢?
Differentiating w.r.t. x.
a((x-a)?+(y-b)?) ac?)
dx ~ dx
a{ (x - a)?) a((y - b)?)
——_——— + —~-0
dx dx

Misc 16 teachoo.com
If cosy =x cos(a + y), withcosa # + 1, prove that
dy _ cos?(at+y)
dx sina
Given
cosy = x cos(a + y)
cosy _
cos(a+ y) =x
_ cosy
x* cos(at+ y)
Differentiating w.r.t.x.
d(x) _ =( cosy )
dx dx cos(a + y)

teachoo.co
Misc 17 aenoo.com
2
Ifx =a(cost + tsint) andy =a (sint - t cos t), Find <=
dy
We need to find —>
ax?
d
First we find “YY
dx
Here,
dy
ay _ dt
dx ax
dt

Misc 18 teachoo.com
if f (x) = [x|?, show that f ”(x) exists for all real x and find it.
We know that
l=} % 228
—x x<0
Therefore,
ype} @ ,x*20
f@=ll hex <0
{x ,x20
—x3 »x<0

Misc 19 teachoo.com
Using the fact that sin(A + B) =sinAcosB +cosAsinB
and the differentiation, obtain the sum formula for cosines.
Given
sin(A + B) = sinAcosB + cosA sinB
Consider A & B are function of x
Differentiating both side w.r.t.x.
d(sin(A+ B)) _ d(sinAcosB +cosA sin B)
dx ~ dx
d(sin(A+ B)) _ d(sinA.cosB) 4 d(cosA.sinB)
dx ~ dx dx
cos (A +B) ; d(A+ B) = d(sin A.cosB) 4 d(cos A.sinB)
dx dx dx

Misc 20 teachoo.com
Does there exist a function which is continuous everywhere but
not differentiable at exactly two points? Justify your answer.
Consider the function
f@) = Ix] +|x—-1]
f is continuous everywhere , but it is not differentiable at
x=O0&x=1
-x-(x-1) x<0
f@)=4x-@-1 O<x<1
x+(x-1) x21
—2x+1 x<0
= 1 O<x<1
2x—-1 x21

Misc 21 (Method 1) feachoo.com
f@) g&) kh) ay [FO gD
lfy=|l om n |,prove that =| 1m n
a b c a b c
a» [FO I@) WO)
Here = = 1 Mm n
ax
a b c
Expanding determinant
dy _\¢ mn os Lon 1 Lom
Za, Ae’ @ll, Ta+ reo, 7]
° = f'(x) (me — bn) — g’(n) (lc — an) + h'(n) (lb — am)
° =(me — bn) f'(x) — (lc — an)g'(x) +(lb — am) h'(x)

Misc 22 teachoo.com
Ify = e@69S"* _ 1 < x < 1,showthat
(1x2) 2% ~~ ® _ ay =0
—x*) + -x—-a@y=0.
dx? dx y
y= et cos "1x
Differentiating w.r. t.x.
dy d(e* cos~1x )
dx . dx
-1
dy _ pacostx y d(acos~+x)
dx dx
dy — pacos tx ( a1 )
ax © *O\ Fe