Limits and Derivatives Class 11

Ex 12.1, 1 teackhoo
Evaluate the Given limit: lim x+3
limx +3
Putting x = 3
=34+3
=6

Ex 12.1, 2 teackhoo
. er 22
Evaluate the Given limit: lim (x = 2)
x7.

: 22
lim (x = =)
XO 7
Putting x ="

=q—2

= 7

teackoo
Ex 12.1, 3
Evaluate the Given limit: lim Tr?
re

lim rr?
rol
Puttingr =1

= 1 (1)

=1

teackoo
Ex 12.1, 4
. a ye AK +3
Evaluate the Given limit: lim ==
x04 K¥—-2
. 4x +3
lim ——_
x04 X-2
Putting x =4
_4(4) +3
~ 4-2
_164+3
~ 2
_19
~ 2

Ex 12.1, 5 teachoo
10 5
Evaluate the Given limit: lim “7***
xo-1 x-1
x Exe HT
lim. ————
KO-1 x-1
Putting x =—-1
_ (40+ C198 41
" (4ajy-4
1-141
“4-1
_O+1
~~ 2
1
~ 2

teackoo
Ex 12.1, 6
SL
Evaluate the Given limit: lim S42
x0 x
5
lim (+1)? -1
x30 x
_ (0+4+1)*-1
~ 0
i -1
ar)
_ 1-1
ar)
=?
a)
ran 0
Since it is of from a
Hence, we simplify

Ex 12.1,7 teackoo
1,
. . as 3K? —K-10
Evaluate the Given limit: lim ——_—
x22 x’ -4
. 3x7-x-10
lim —W——_
x02 x" -4
_ 3x27-2-10
~ 22-4
_3x4-2-10
“44
_ 12-12
“4-4
_9
a)
Since it is of form a

teackoo
Ex 12.1, 8
A aaa ye 4-81
Evaluate the Given limit: lim ————
x33 2x° —5x-3

lim —7 —*4
x53 2x? -5x-3
Putting x = 3

_ (3)*-81

~ 23) -5()-3

_ 81-81

“48-15-3

_o

a)

. a 0 ar

Since it is a 3 form, we simplify as

Ex 12.1,9 teachoo
Evaluate the Given limit: lim axed
x30 cxt1
. axtb
lim ——
xo0 cx +1
Putting x =0
_a(0) +b
~ ¢(0) +1
_O+b
“O+F1
_b
“4
=b

Ex 12.1, 10 teackoo
1
. a 2871
Evaluate the Given limit: lim —=——
zol 76-1
1
. Z3-1
lim TT
zZo1 26-1
1
(13-1
=a
(1)6 -1
_i-41
“4-1
a)
a)
ag 0
Since it is form 0

teackoo
Ex 12.1, 11 a
2
Evaluate the Given limit: lim oes atb+c#+0
x1 cx’+bx+a
. ax? t+bxtc

lim ———
x1 cx*+bx+a
Putting x =1

_ a(1)? +b) +

~ ¢(4)? + ba) +a

_atbte

“atbte

=1

Ex 12.1, 12 teackoo
Evaluate the Given limit: lim. —
x7-2xKX+2
lim. +
x2-2x+2
. . 0
At x =—2, the value of the given function takes the form .
So, simplifying
1 1 2Z+x
a42 2+x
lim ¥~= lim Ge) x)
x2-2xX+2 x35-2 x+2
. 1
= lim —
x72 2x
Putting x = —2
1
~2(-2)
_ -1
~ 4

Ex 12.1, 13 teackhoo
Evaluate the Given limit: lim ——~
x0 ~bx
. sin ax
lim ——
x20 bx
. . 1
=lim sin ax x—
x0 bx
Multiplying & dividing by ax
. . 1 ax
=lim sin ax x— x—
x0 bx ax
. sinax ax
= lim —— x—
x20 ax bx
. sinax a
= lim —— x—
x20 ax b
a . sinax
=— x lim —
b x-0 ax

Ex 12.1, 14 teackoo
Evaluate the Given limit: lim a a,b #0
x 0 sin bx
li sin ax
sim sin bx
=lim sin ax x lim —
x0 x0 sin bx
Multiplying & dividing by ax
. sin ax . ax
= lim — x lim —
x30 ax x 0 sin bx
=1x lim—_
x0 sin bx sin x
Using lim —— =1
= lim mos
~ x00 sin bx :
Replacing x by ax
Multiplying & dividing by bx lim sin ax _ 1
x30 ax
=li ax, bx
~ ime sin bx bx

Ex 12.1, 15 teackhoo
Evaluate the Given limit: lim “2 2—
xom (TT — x)
lim sin (™ — x)
xOnm (1 — xX)
Lety=m —-x
So, when x > 1
yorn-n
y>o
So, our equation becomes
.__ Sin (1 —x) . siny
lim —— =lim—
xn W(t —x) yoo ny

Ex 12.1, 16 teachoo
. ge COS XK
Evaluate the given limit: lim ——
x70 T —-xX
. COSX
lim —
x70 1 —X
Putting x =0
_ cos 0
“#-0
_ cos 0
ae:
_1
— (cos0 = 1)

Ex 12.1, 17 teackoo
. a 2x1
Evaluate the Given limit: lim <<~—
x00 cosx-1
. cos2x—1
lim ————_
x70 cosx-1
: 2eos?x-1)-1 .
= lim Goo zaU=" (Using cos 2x = 2cos? x - 1)
x00 cosx —
. 2eos?x-1-1
= lim —W——
x0 cosx-1
. 2c0s*x —2
= lim —————_
x30 cosx-1
. 2(cos?x-1 . .
= lim 2(cos” x— 1) Using sin? x = 1 —cos? x
0 1 9
x90 cosx—
._ 2(cos? x — 1?
= lim 2608 = 1)
x90 cosx-1

Ex 12.1, 18 teachoo
Evaluate the Given limit: lim ==?" *
x00 ~bsinx
... ax+xcosx
lim —————_
x90. = =bsinx
- X (a+ cos x)
= lim ——_——
x00 b sinx
= lim oe“ x x )
x70 b sin x
: a+ cos x) sin x
x0 b x
. at+cosx . sinx
=lim (eo) + (lim =)
x70 b x0 x
. a+cosx i
= lim Ss) +1 (Using lim “2% = 1)
x30 b x00 x

Ex 12.1, 19 teackhoo
Evaluate the Given limit: lim x sec x
x00
lim x sec x
x70
=limx. — i a
~ ¥50 * cos x (Using secO=— =)
. x
=lim —
x-0 cosx
Putting x=0
an)
~ cos 0
0
= - cosO=1
7 ( )
=0

Ex 12.1, 20 teachoo
Evaluate the Given limit: lim Smanyhe a,b,a+b+#0
x00 ax + sin bx
.. Sinax + bx
lim ————
x30 ax + sin bx
‘sin ax
x50 x(a + sin >)
x
‘sin ax
= lim Ce )+e x ) *°
sin bx
a (BP)
Multiply & Divide by == by ax
& Multiply & Divide _— by bx
sinax ax
=lim ( x a) +b
sin bx bx
x0 a +( x a)

Ex 12,1, 21 (Method 1) teackoo
Evaluate the Given limit: lim (cosec x — cot x)
x7
lim (cosec x — cot x)
x70
: 1 cos x : _ 1
=lim (= - <*) Using cosec 8 =—— @
x30 \sinx sin x sm
cot 6 = cos
. 1-cosx sin @
= lim ————_
x00 sinx
Putting x=0
_ 1-—cos0O
~ sind
_i-1
~ 0
_ 0
a)

Ex 12.1, 22 teackhoo
. tan 2x
lim. —>+
xo Et x- >
2 2
: tan 2x
lim ——
xO E-F
2 2
Putting y =x -
When x > 5
5 =_=
Yr a3
y>o
So, our equation becomes

Ex 12.1, 23 (Method 1} teackoo
Find lim f(x) and lim f(x), where f(x)
x00 xo1
_f 2x43. x <0
~)3(%+1), x>0
Finding limit at x = 0
Hing Fo) = Jin 0) = Ta
LHLatx > 0 RHL at x > 0
ign 9 Li 0
= lim (2x + 3) = lim 3(x +1)
= 2(0)+3 = 3(0+1)
=0+3 = 3(1)
=3 =3

teackoo
Ex 12.1, 24 (Method 1)
ae _fx—-1, x <1
Find lim f(x), where f(x) = {2 -1,x>1
The Limit at x = 1 will be
dim fad = Jim. fx) = ir, fo)
LHLatx>1 RHLatx->1
Hint dirt
=li 28 = li -x2-
jim x? -1 lim (-x? - 1)
=(1)°-1 =-(1)-1
=1-1 =-1-1
=0 =-2
Thus, lim, f(x) 4 lim f(x)

Ex 12.1, 25 teackoo
tl x #0
Evaluate lim f(x), where f(x) =4 x ,
x30 0, x=0
Finding limit at x = 0
LHL atx -0 RHL atx ~0
Jim f(x) = lim f(O- h) Jim, f(x) = lim f(0 + h)
= fing) = fing
ERI _ lal
= lim —_ = lim —
ho-0 —h ho0 A
h h
= lim — =lim—
h>0 —h hoo h
=lim-1 =lim1
hoo h>0
=-1 =1

Ex 12.1, 26 teackoo
aT 0
Evaluate lim f(x), where f(x) = fs, xX#
x0 0, x=0
Finding limit at x = 0
LHL atx > 0 RHL atx >0
Jim. f(x) = lim f(0- h) Jim, f(x) = lim f(0 + h)}
= lim {(-h) = lim f(h)
. Th A
= lim — = lim —
h-0 |—-hAl h>0 [A]
—h h
= lim — =lim—
ho0 h hoo h
= lim -1 =lim1
h>o h-0
=-1 =1

Ex 12.1, 27 teackoo
Find lim f(x), where f(x) = |x| —5
xO.
Finding limit at x = 5
LHLatx—> 5 RHLatx>5
Jim. f(x) = lim #(5 -h) Jim, f(x) = lim f(5 +h)
= lim [5 —hA|-5 =lim |[5+A|-5
h30 hoo
=lim(S—h)-5 =lim(S+h)-5
=lim—h =limh
h30 h>0
=0 =0
Thus, lim f(x) = 0
x05

teackoo
Ex 12.1, 28
at+tbx, x<1
Suppose f(x) =< 4, xX = 1 and if lim f(x) = f(1) what are
b-—ax, x>1 xl
possible values of a and b?
Here, limit exist at x > 1
ie., LHL = RHL = f(1) = 4 (1)
LHLatx->1 RHLatx->1
jim. f(x) = lim f(1-h) jim, f(x) = lim f(1 +h)
=lima+b(1-h) = lim b-a(1+h)
h30 h50
=a+b(1-0) =b-a(1+0)
=atb (2) =b-a ...(3)

Ex 12.1, 29 teachoo
Let aj, a,....., a, be fixed real numbers and define a function
F(x) = (x — ay) (X— ap)... (X- ay)
What is lim f(x)? For some a # aj, ap......a,, compute lim f(x).
XPay xv-a
f(x) = (x — a4) (X — a5) «(x — ay)
Calculating lim f(x)
x7 ay
lim f(x) = lim (x-a,) (x-a))..... (k-a,)
Xa, Xa,
Putting x= a,
= (a, — a,) (a, — a9) .....(€)— ay)
=0x (a,-4a))...... (a, -—a,)
=0

Ex 12.1, 30 teackhoo
Ix] +1, x<0
If f(x) = 4 0 x=0.
Ix] -1, x>0
For what value (s) of a does lim ffx) exists?
xra
We need to find value of a for which lim f(x) exists
xX-~a
We check limit different values of a
* Whena=0
* Whena<0O
* Whena>O
Case 1: When a=0

Ex 12.1, 31 teachoo
; toe - f(x)-2 .
If the function f(x) satisfies lim —~— =, evaluate lim f(x).
xo. x -1 xo
Given
. -2
lim [@)~? =
x71 x*4-1
By Algebra of limits
lim £2). - 28a
ea g(x) lim g(x)
xoag pm g
lim f@) 2 _
lmo2-1
xOL
lim (f(x) - 2) = nx Lim (x? - 1)
xo1 xol
lim f(x) — lim 2 = n (lim x? - lim 1}
xol1 xol1 xo1 xol1

Ex 12.1, 32 teachoo
mx? +n, x<0
If f(x) = 4 nx +m 0<x< 1. For what integers mand n
nx? + m, x>1
does lim f(x) and lim f(x) exist?
x00 xo1
Given limit exists at x =O and x=1
Atx=0
Limit exists at x = 0 if
Left hand limit = Right hand limit

Ex 12.2, 1 teackoo
Find the derivative of x? -2 atx = 10.
Let f (x)= x7-2
py — x? — 2)
P(x) =~
=2x-0
=2x
So, f’(x) = 2x
f’(10) = 2x 10
=20

Ex 13.2, 2 teachoo.com
Find the derivative of x at x = 1.
Let f (x) =x
We need to find derivative of f(x) at x = 1
ie. f’ (1)
We know that
. h)—
(9 = lim Le EW ALO
h70 h
Here, f(x} = x
So, f(x+h)=x+h

Ex 12.2, 3 teachoo
Find the derivative of 99x at x = 100
Let f (x) =x
We need to find derivative of f(x) at x = 100
i.e. f’ (100)
We know that
. +h)-
f’ (x) = lim£& ) — f )
ho0 h

Here, f(x) = 99x

f(x + h) = 99(x + h) = 99x + 99h

Ex 12.2, 4 teachoo
Find the derivative of the following functions from first principle.
(i) 8-27
Let f(x) =x? - 27
We need to find Derivative of f(x)
ie. f” (x)
We know that

” _ Lee (OK +h) - FC”)

Fea) = fig
Here, f (x) = x? -— 27

f(x +h) =(x+ h)?-27

Find the derivative of the following functions from first principle.
(ii) (x – 1) (x – 2)

Find the derivative of the following functions from first principle.
(iii) 1/𝑥^2

Find the derivative of the following functions from first principle.
(iv) (𝑥 + 1)/(𝑥 − 1)

Ex 12.2, 5 teachoo
; ery 2
For the function f(x} = — + — +....4—+x+ 1. Prove that f’(1} = 100
100 = =©100 2
f(0)
We have
100 99 x2 (x")’ = nxe-t
f (x) =——+— +... + —4+x41 ,
100 99 2 & {a)’=0
where a is constant
1 1 1 '
f* (x) = (= x1004 — X99 HS AE XE 1)
100 99 2
y 1 100-1 1 99-1 1 2-1 1-1
f’ (x) = — x 100x + — x 99x +. b= & 2x2-14 1.1140
100 99 2
100 99 2
= — x99 + — x98 4 t-te x0 40
100 99 2
= x99 4 x94 Xt 140
= x99 498 + +x +1

Ex 12.2, 6 teackhoo
Find the derivative of x" + ax"? + a2x"+....+ a" x + a" for some fixed
real number a.
Let f (x) = x" + ax"-1+4 a2x"~24,,..4 a? 1x + a" (where a is constant)
(x?)’=nx?-2
f? (x) = (xP + axn tt a2xh2ht an 1X + an)’ & (a)’=0
where a is constant
=n.x" 14 a(n — px 9-14 a2 (mn — 2px ot.
+a"? (1pco1+0
= nx? t+ a(n — 1)x?-2 + a2 (n—2)x 9-3 +... ah 4 (1)(1)
=nx""14a(n—1)x"-2 4. a2 (n—-2)x "3 4... 4¢an-2
of) = nx" 1 4 a(n —1)x"-2 + a2 (n—-2)x "3 + tant

Ex 12.2, 7 (Method 1) teachoo
For some constants a and b, find the derivative of
(i) (x— a) (x—b)
Let f (x) = (x-a) (x—b)
=x(x-b)-a(x-b)
= x?—xb—ax tab
=x?—(b+a)x + ab
Oey = nxr-t
f’(x) = (x? - (b +. a)x + ab)’ & (a)’=0
=2.x21-(b+a).Dd1+0
= 2xt-(b +a). x°
=2x-(bt+a)x1
= 2x—-(b+a)

For some constants a and b, find the derivative of
(ii) (ax2 + b)2

For some constants a and b, find the derivative of
(iii) (x − a)﷮(x − b)﷯

Ex 12.2, 8 teachoo
n_ an

Find the derivative of — for some constant a.
We need to find derivative of —

x "— a?
Let f(x) = yea
Let u=x"-a"&v=x-a

u
So, f(x) = (“)
0, F(x) = {—
fay aol
f(x) = u 7 u

Ex 12.2, 9 teackhoo
Find the derivative of
(i) 2x - =
3
Let f(x) = 2x -=
4
, d(2x -)
f(x) = —
= 2-0
=2
2 f(xj=2

Find the derivative of
(ii) (5x3 + 3x – 1) (x – 1)

Find the derivative of
(iii) x–3 (5 + 3x)

Find the derivative of
(iv) x5 (3 − 6x−9 ).

Find the derivative of
(v) f (x) = x–4 (3 – 4x–5)

Find the derivative of
(vi) f(x) = 2/(x + 1) – x2/(3x − 1)

Ex 12.2, 10 teachoo
Find the derivative of cos x from first principle.
Let f (x)= cos x
We need to find f’(x)
We know that
. h)-
f(x) = lim fat h)- f@)
hoo h
Here, f (x) = cos x
f (x + h) = cos (x +h)
Putting values,

Ex 12.2, 11 teackhoo
Find the derivative of the following functions:
(i) sin x cos x
Let f (x) = sin x cos x.
Let u=sinx

&v=cosx
« f(x) = uv
So, f(x} = (uv)’

=uvtv'u
Here, u=sinx
u’ = cos x (Derivative of sin x = cos x)

Find the derivative of the following functions:
(ii) sec x

Find the derivative of the following functions:
(iii) 5 sec x + 4 cos x

Find the derivative of the following functions:
(iv) cosec x

Find the derivative of the following functions:
(v) f (x) = 3cot x + 5cosec x.

Find the derivative of the following functions:
(vi) f (x) = 5sin x – 6 cos x + 7.

Find the derivative of the following functions:
(vii) f (x) = 2 tan x – 7 sec x

Example 1 teachoo.com

Find the limits:

(i) Lim [x3 — x2 + 1]

x1

lim [x3 — x? + 1]

xXOL

Putting x = 1
=(1P-(1P +1
=1-1+1
=0O+1
=1

Example 1
Find the limits:
(ii) (𝑙𝑖𝑚)┬(𝑥→3)⁡〖 [𝑥(𝑥+1)]〗

Example 1
Find the limits:
(iii) (𝑙𝑖𝑚)┬(𝑥→−1)⁡〖[1+𝑥+𝑥2+…..+𝑥10]〗

Example 2 teachoo.com
Find the limits:
ype 241
(i) Lim [|
x1 Lx + 100

lim [+]
xo1 Lx + 100.
Putting x =1

_ vet

~1+100

_att

~ 401

2.

~ 101

Example 2
Find the limits:
(ii) (𝑙𝑖𝑚)┬(𝑥→2) [(𝑥^3 − 4𝑥^2 + 4𝑥)/(𝑥^2 − 4)]

Example 2
Find the limits:
(iii) lim┬(x→2) [(x2 −4)/(x3 − 4x2 + 4x)]

Example 2
Find the limits:
(iv) lim┬(x→2) [(x3 −2𝑥2)/(x2−5x+6)]

Example 2
Find the limits:
(v) lim┬(x→1) [(x −2)/(x2−x)− 1/(𝑥3 −3𝑥2+2𝑥)]

teackoo.com
Example 3
Evaluate:
xi
i) lim =
ti xot xt? 4
. x4
lim arr
xo1 x -1
_ ay -1
~ (1)10 -4
_1-1
“4-4
2
“0
a 0
Since it is form .

Example 3
Evaluate:
(ii) (𝑙𝑖𝑚)┬(𝑥→0) (√(1 + 𝑥) − 1)/𝑥

teachoo.com
Example, 4
Evaluate:
+y ys... SiN 4x
(i) lim ——
x00 Sin 2x
.. sin 4x
lim —_!
x 0 sin 2x
= lim sin 4x x lim ——
x00 x00 sin 2x
Multiplying & dividing by 4x
. . Ax + 1
= lim sin 4x .— x lim ——
x00 Ax x00 sin 2x
.. sin4x : 1
= lim ——. 4x x lim ——
x00 4x x00 sin 2x
. sin 4x . . 1
= lim —— x lim 4x x lim ——
x00 4x x70 x00 sin 2x
1 Using lim asi
. . QO x
=1x lim 4x x lim —— ~~
x0 x30 sin 2x Replacing x by 4x
in 4x
lim 2 = 3
x30 4x

Example 4
Evaluate:
(ii) lim﷮x→0﷯ tan﷮x﷯﷮x﷯

teachoo.com
Example 5
Find the derivative at x = 2 of the function f(x) = 3x.
f (x) = 3x
We know that,
’ ye, fx+h)-f(x)
f’(x) = lim —
Now, f(x) = 3x
So, f(x + h} = 3 (x +h}
’ _ yn, 30 +h) - 3 &)
f’ (x) = lim Ss
Putting x = 2
pray, 3(2 +h) -3 (2)
f’ (2) = lim —_

feachoo.com
Example 6 (Method 1)
Find the derivative of the function f(x) = 2x2+ 3x-5 atx= -1.
Also prove that f’(0) + 3f’(—1) =0.
Given f(x) = 2x2 + 3x-5
We know that
’ . f(x +An)—-f(x)

f’(x) = lim ————>
Now f (x) = 2x? + 3x-5
So, f (x +h) = 2(x +h)? + 3(x+h)-5
Putting values

2 - - 2 =-

f’ (x) =lim (2(¢ +h)? + B(x +h) - 5) — (2x2 4+ 3x -5)

h-0 h

Example 7 teachoo.com
Find the derivative of sin x at x =0.
Let f(x) = sin x
We know that
He) = Tin, Ft) - FO)
i
Here,
f(x) = sin x
f(x + h) = sin (x + h)
Now,

Example 8 teachoo.com
Find the derivative of f(x) = 3 at x = O and at x =3.
f(x) = 3
We need to find
Derivative of f(x) atx =0 & atx =3
i.e. f? (0) & f’ (3)
We know that

f(x) =lim f(x +h) - f(x)

hoo h

Here, f (x) =3
So, f (x +h} =3
Putting values

“1y) = Jim 222

rs) = m5

Example 9 teachoo.com
Find the derivative of f(x} = 10x.
Let f (x) = 10x
We need to find derivative of f(x)
ie. f’ (x)
We know that

f(x) = lim f(x +h) - f(x)

hoo h
Here, f {x} = 10x
So, f (x + h) = 10({x + h)
Putting values
f’ (x) = lim 10(x +h) -10x
ho0 h

feachoo.com
Example 10
Find the derivative of f(x) = x2.
Given f(x) = x?
We need to find derivative of f(x)
ie. f? (x)
We know that
f’(x) = lim f(x +h) — fG)

h-0 h
Here, f (x) = x?
So, f (x +h) = (x +h)?
Putting values

teachoo.
Example 11 faemoo ton
Find the derivative of the constant function f (x) = a for a fixed real
number a.
Given f (x) =a
»where ais constant
thus, f(x) is a constant function
We need to find derivative of f(x)
i.e. f’ (x)
We know that
, . f(x +h) - f()
f(x) = Jim ————>
(x) = jim h
Here, f {x}=a
So, f (x+h)=a

Example 12 teachoo.com
Find the derivative of f(x) =*
1
f (x)= =
We need to find Derivative of f(x)
ie. f (x)
We know that
f(x) = lim f(x +h) - f(x)
h-0 h
1
Here, f (x) = :
So, f (x +h) = a
(x +h)

Putting values

teachoo.com

Example 13
Compute the derivative of 6x1 — x55 + x,
Let f (x) = 6x100 — x95 + x
We need to find f’ {x}
#7 (x) = 6 (100) x199-1— 5555-144, xt-1 (As (x°)’ = nx°-4)

= 600x°9 — 55x 54+ x?

= 600x°? — 55x54 + 1

feachoo.com
Example 14
Find the derivative of f(x) =1+x+x? 4x3 +..4x°% at x= 1.
We need to find f’ {x} at x = 1
ie. f’ (1)
f(x) = 14+ xX4x7 4x3 to, OP
f(x) = (1 tx x72 $8 to # OO) Oey = nxr-t
f’(x) = OF D.xbt + 2x24 4 3x32 tose + 50X01 & (a)'=0
where a is constant
= 1+ 2x + 3x? tu. + 50X49
Hence,
f (x) =14+ 2x + 3x2 +... + 50x

Example 15 (Method 1) feachoocom
Find the derivative of f(x) = =
x+1
f(x) = 224
x 1
on
x x
=1+ 1
x
=1+(x)*
Now,
f’ (x) = (1 + (x) 7)
(x) = (1+ 00) ey oma
=0+(-1)x7>1 & (a)’=0
= 0-x where a is constant
= —-x?
_-1
we

Example 16 teachoo.com
Compute the derivative of sin x.
Let f (xj = sin x
We need to find f’(x)
We know that
, f(x +h) - f(x)
f’(x) = lim ————>
Here, f (x) = sin x
So, f (x +h) = sin (x +h)
Putting values
#’(x) = lim St = sin x
x) noo h

Example 17 teackoo.com
Compute the derivative tan x.
Let f(x) = tan x
We need to fina f’ {x)
We know that
f’(x) =lim f(x + h) — £ (x)
h>0 Rh
Here,
f(x) = tan x
f(x + A) =tan (x +h)
Putting values

feachoo.com
Example 18
Compute the derivative of f(x) = sin? x.
Let f(x) = sin? x
f(x) = sin x sin x

Let u=sinx &v=sin x
So, f(x) = uv
Now,

(x) = (uvy’
Using Product rule

f(x) =u’v + vu

Example 19 teachoo.com
Find the derivative of f from the first principle, where f is given by
. 2x +3
(i) fq =
We need to find Derivative of f(x)
ie. f’ (x)
We know that
Hy) = Hin FAM F@)
F(x) = im SS
2x4+3
Here, f(x) =o
2 +h) +3
So, f (x +h) = G2

Example 19
Find the derivative of f from the first principle, where f is given by
(ii) f(x) = x + 1/x

Example 20 teackoo.com
Find the derivative of f(x) from the first principle, where f(x) is
(i) sin x + cos x
Given f (x) = sin x + cos x
We need to find Derivative of f(x)
We know that
h =
) = lim LEED =F)
h>0 h

Here,

f (x) = sin x + cos x

f (x +h) =sin (x +h) + cos (x + h)

Example 20 teachoo.com
Find the derivative of f(x) from the first principle, where f(x) is
(ii) x sin x
Given f (x) =x sin x
We need to find Derivative of f(x)
We know that
h =
FG) = lim LEED = FO)
h>0 h
Here, f (x) =x sin x
So, f (x+h}= (x +h) sin (x +h)
Putting values

teachoo.

Example 21 eachoo.com
Compute derivative of
(i) f(x) = sin 2x
Let f (x} = sin 2x

= 2 sin x cos x
Let u=2sinx &v=cosx
So, f(x) = uv
+ #'(x) = (uv)’

=uvetv'u

Example 21
Compute derivative of
(ii) g(x) = cot x

Example 22 teackoo.com
Find the derivative of
. x5 —-cosx
(i) sin x
x? —-cosx
Let f(x) = —————
sinx
Let u=x°-—cosx &v=sinx
uu
So, f(x) = ()
w=)
o xp= 7"
Using quotient rule

Example 22
Find the derivative of
(ii) (𝑥 + 𝑐𝑜𝑠⁡𝑥)/𝑡𝑎𝑛⁡𝑥

Misc 1 teachoo.com
Find the derivative of the following functions from first principle:
(i) -x
Let f (xj =-—x
We need to find derivative of f(x}
ie. f’ (x}
We know that

f(x) = lim fGcth)- fF)

ho h

Here, f (x) =—x
So, f (x +h) =— (x +h)

Misc 1
Find the derivative of the following functions from first principle:
(ii) (−𝑥)^(−1)

Misc 1
Find the derivative of the following functions from first principle:
(iii) sin (x + 1)

Misc 1
Find the derivative of the following functions from first principle:
(iv) cos (x−π/8)

Misc 2 feachoo.com
Find the derivative of the following functions (it is to be
understood that a, b, c, d, p, q, rand s are fixed non-zero
constants and m and n are integers):
(x +a)
Let f(x) = (x + a)
(xty’ = nx? -1
& (a)’=0
where a is constant
f? (x) = 1x1-1+0
=x?
=1

Misc 3 (Method 1) teachoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n
are integers):
(px + q) ¢ + s)
Let f(x) = (px + q) ¢ + s)
= (px + q) (rx “++ s)
Letu=(pxt+q) &v=(rmt+s)
- f(x) = uv
So, f’(x) = (uvy’
F(x) = uv+ vu Using product rule,
(uv)’ = u’v+v/u

Misc 4 teachoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and
n are integers):
(ax + b) (cx +d)?
Let f(x) = (ax + b) (cx + d)?
Letu=ax+b &v=(cx+d}
So, f(x) = uv
f(x) = (uvy’
=uvtv'u

feachoo.com
Misc 5
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and
n are integers): axth
cxtd
Let f(x) = 2 +2
cxt+d
Letu=axtb&v=cx+d
u
ia
“yy = (4%
So, f(x) = (-)
, ulv—vlu
ra

Misc 6 teackoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n
are integers):
142
x
1 = 1
x
1+2
Let f(x) = 72
x
x+1
__x
= x-1
x
x1 x
=— xX —
x x-1
_xti1
“x4

Misc 7 feachoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and
. 1
n are integers): eabeae
1
Let f(x} = ax’ +bx+c
Letu=1&v=ax? +bx+c
u
ia
yy = (%
So, f(x) = (-)
, ulv—vlu
F(x) = =

Misc 8 teackoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and
n are integers):
ax+b
px’+qxtr
ax+b

Let f(x) = —>———_

pxt+qx+r
Letu=ax+b&v=px? +qx+r

u
f
yy a (Ut

So, f(x) = (¢)

Misc 9 teackoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and
n are integers):
px’+qxtr
ax+b
r+ gx+r
Let f(x) = 22 ="
ax+b
Let u=px?+qx+r&v=axt+b
u

ie u '

So, f(x) = (¢)

: teachoo.
Misc 10 Paenoo conn
Find the derivative of the following functions (it is to be
understood that a, b, c, d, p, q, rand s are fixed non-zero constants

. b
and m and n are integers): 5 — zat COS X
a b
Let f(x) = ya yet cos x
f(x) = ax-4—bx-? + cos x
f’(x) = (ax -4— bx -2 + cos x)’
= (ax —4)’ — (bx ~2)’ + (cos x)’

Derivative of x" is nx"-1&

Derivative of cos x = —sinx
=a.(-—4)x-4-1 -—b. (-2)x727-14(-sinx)
=—-4ax->+ 2bx-?- sin x
_-—4a 2b .
= +Z ~sinx

Misc 11 feackoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n
are integers): 4./x -2
Let f(x) = 4/x-2
i
= A(x)2 — 2
i
F(x) = (4(x)2 — 2
(x?) = nxt-t
1, 4-1
= 4.5 (x)? -0 & (a)’=0
a-2 where a is constant
= 2x 2
at
= 2x 2
2
a7
(2
=2
VE

Misc 12 (Method 1) feachoo.com
Find the derivative of the following functions (it is to be
understood that a, b, c, d, p, g, rand s are fixed non-zero
constants and m and n are integers): (ax + b}"

Let f(x) = (ax + b)".
We know that

, =}; f(x + h) - f(x)
=m SO =

Here, f(x) = {ax + b)"
So, f(x + h) = (a(x + h) + b)"

Putting values

f(x) = lim (a(x+h)+b)” — (ax + by"

hoo h

feachoo.com

Misc 13
Find the derivative of the following functions (it is to be
understood that a, b, c, d, p, g, rand s are fixed non-zero
constants and m and n are integers): (ax + b)" (cx + d)™
Let f(x) = (ax + b)" (cx +d)"
Let u=(ax+b)"&v=(cx+d)™
f(x) = uv
So, f’ (x) = (uv)’

f’(x) = u’v + v’u

Misc 14 teachoo.com
Find the derivative of the following functions (it is to be
understood that a, b, c, d, p, q, rand s are fixed non-zero constants
and m and nare integers): sin (x + a}
Let f(x) = sin (x +a)
Using sin (A + B) = sinAcos B +cosB sinA
=sinx.cosa +cosx.sina
= cos a (sin x) + sin a (cos x}
So, f’(x) = (cos a (sin x) + sin a (cos x))’
= (cos a (sin x))’ + (sin a (cos x))’
=cos a (sin x)’+sina(cosx) (cosa & sina is constant)
=cosacosx+sina(—sinx) Derivative of sin x = cos x
Derivative of cos x =—sinx
= cos acos x—sinasinx

. feachoo.com
Misc 15
Find the derivative of the following functions (it is to be
understood that a, b, c, d, p, g, rand s are fixed non-zero constants
and m and n are integers):

cosec x cot x

Let f(x) = cosec x cot x
Let u=cosecx &v=cotx
So, f(x) = uv
F(x) = (uv)’
Using product rule
f’(x) = u’v+v’u

. teachoo.
Misc 16 Paenoo conn
Find the derivative of the following functions (it is to be
understood that a, b, c, d, p, q, rand s are fixed non-zero constants
and m and n are integers): Se

1+sin x
Let f (x) = ——*
1+sinx
Let u=cosx &v=1+sinx
u
So, f(x) = >
oa u '
Using quotient rule
’ ulv—vlu
(x) =

Misc 17 teackoo.com
Find the derivative of the following functions (it is to be
understood that a, b, c, d, p, q, rand s are fixed non-zero constants
and m and n are integers): SDK COS
sin x—cosx
sin x + cos x
Let f (x} = ————————
sin X— cosSx
Letu=sinx+cosx & v=sinx—cosx
u
UL f
So, f’(x) = ()
Using quotient rule

Misc 18 teackoo.com
Find the derivative of the following functions (it is to be
understood that a, b, c, d, p, q, rand s are fixed non-zero constants
and m and n are integers): secxat
secx+1
secx—-1
Let f (x} = ———_
secxt+1l
Letu=secx-—1&v=secx+1
UuU
CU ta
So, f’(x) = ()
Using quotient rule

feachoo.com

Misc, 19
Find the derivative of the following functions (it is to be
understood that a, b, c, d, p, g, rand s are fixed non-zero constants
and m and n are integers): sin" x
Let f(x) = sin" x.
Let p =sinx
So,f(x) =p”
By Leibnitz product rule
f(x) = (p"y’ p’

=npr tp’
Putting p = sin x

=nsin"~1x (sin x)’

. teachoo.
Misc 20 Paenoo conn
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n

int ): at+bsinx
are integers): —
at+bsinx
Let f (x) ~~ c+dcosx
Letu=a+bsinx &v=c+dcosx
uu
So, f(x) = ()
- u '
Using quotient rule
, ulv—v'u
P(x) =

Misc 21 feackoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n
are integers): sins + a)
cosx
Let f (x) = sin(x + a)
cosx
Let u=sin(x+a) &v=cosx
UU
U f
So, f(x) = ()
Using quotient rule
, ulv—v'u
P(x) =

Misc 22 feachoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n
are integers): x* (5 sin x — 3 cos x)
Let f (x) = x* (5 sin x-— 3 cos x)
Let u=x* &v=5sinx-—3cosx
+ f(x) = uv
So, f’(x) = (uv)!
f'(x)=u'v — vu
Finding uw’ & v’

u=xt

u’ = 4x4-1 (As (x")’ =n x1)

= 43

feachoo.com

Misc 23
Find the derivative of the following functions (it is to be
understood that a, b, c, d, p, q, rand s are fixed non-zero constants
and m and n are integers): (x + 1) cos x
Let f (x) = 0 +1) cos x
Let u ={x?+ 1) &v=cosx
o f(x} = uv
So, f’(x) = (uv)!
Using product rule

f(x)su'v+ v'u

Misc 24 teachoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n
are integers):
(ax? + sin x}( p + q cos x)
Let f (x) = (ax? + sin x) (p + q cos x)
Let u=ax’+sinx &v=p+qcosx
- f(x) = uv
So, f’(x) = (uvy’
Using product rule
=u'vt vu

Misc 25 feachoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n
are integers):
(x + cos x) (x — tan x}
Let f (xj = (x + cos x) (x — tan x)
Let u=x+cosx &v=x-tanx
+ f(x) = uv
So, f’(x) = (uv)’
Using product rule
f'(xj=ulv t+ vu
Finding u’ & v’

. feachoo.com
Misc 26
Find the derivative of the following functions (it is to be
understood that a, b, c, d, p, q, rand s are fixed non-zero constants
+ 4x+5sinx
and m and n are integers): ————
3x +7 cosx
4x+5sinx
Let f (x) ~ 3x +7 cosx
Letu=4x+5sin &v=3x+7cosx
u
oa u '
So, f (x) = ()
Using quotient rule
, ulv—vlu
P(x) =

Misc 27 teackoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and
. x? cos

n are integers): x.

x? cos=
Let f (x) = ——+

sinx
ris .

Let u =x’ cos > &v=sinx

Uu
So, f(x) = >

Ue t
Using quotient rule

Misc 28 teackoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and
. t ): x

nare integers): ——
Let f (x) = —~—

1+tanx
Letu=x &v=1+tanx

Uu
So, f(x) = >

Ue t
Using quotient rule

Misc 29 teackoo.com
Find the derivative of the following functions (it is to be
understood that a, b, c, d, p, q, rand s are fixed non-zero constants
and m and n are integers): (x + sec x) (x — tan x)
Let f(x) = (x + sec x) (x — tan x)
Let u=x+secx &v=x-tanx
- f(x) = uv
So, f’(x) = (uvy’
Using product rule
f(x)=ul'v t+ v'u

Misc 30 teackoo.com
Find the derivative of the following functions (it is to be understood
that a, b, c, d, p, q, rand s are fixed non-zero constants and m and
. x
n are integers): wns
x
Let f(x) = ——
stn x
Letu=x&v=sin"x
Uu
U f
so, r)=(")
0, Fx) = (5
Using quotient rule