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Ex 12.2

Ex 12.2, 1

Ex 12.2, 2

Ex 12.2, 3

Ex 12.2, 4 (i) Important

Ex 12.2, 4 (ii)

Ex 12.2, 4 (iii) Important

Ex 12.2, 4 (iv)

Ex 12.2, 5

Ex 12.2, 6

Ex 12.2, 7 (i) Important

Ex 12.2, 7 (ii)

Ex 12.2, 7 (iii) Important

Ex 12.2, 8

Ex 12.2, 9 (i)

Ex 12.2, 9 (ii) Important

Ex 12.2, 9 (iii)

Ex 12.2, 9 (iv) Important

Ex 12.2, 9 (v)

Ex 12.2, 9 (vi) You are here

Ex 12.2, 10 Important

Ex 12.2, 11 (i)

Ex 12.2, 11 (ii) Important

Ex 12.2, 11 (iii) Important

Ex 12.2, 11 (iv)

Ex 12.2, 11 (v) Important

Ex 12.2, 11 (vi)

Ex 12.2, 11 (vii) Important

Last updated at May 29, 2023 by Teachoo

Ex 12.2, 9 Find the derivative of (vi) f(x) = 2/(x + 1) – x2/(3x − 1) Let f (x) = 2/(x + 1) – x2/(3x − 1) Let f1 (x) = 2/(x + 1) & f2 (x) = x2/(3x − 1) ∴ f(x) = f1(x) – f2 (x) So, f’(x) = (f1(x) – f2(x))’ f’(x) = f’1(x) – f’2(x) Finding f1‘(x) f1 (x) = 2/(𝑥 + 1) Let u = 2 & v = x + 1 ∴ f1(x) = 𝑢/𝑣 Now, f1’(x) = (𝑢/𝑣)^′ f1’(x) = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 u = 2 u’ = 0 v = x + 1 v’ = 1 + 0 = 1 f’1(x) = (𝑢/𝑣)^′ = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 = (0(𝑥 + 1) −1 (2))/(𝑥 + 1)2 = (−2)/〖(𝑥 + 1)〗^2 Hence, f1’ (x) = (−2)/(𝑥 + 1)2 Finding f2‘(x) f2 (x) = 𝑥2/(3𝑥 − 1) Let u = x2 & v = 3x – 1 Now, f2’(x) = (𝑢/𝑣)^′ f2’(x) = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 Finding u’ & v’ u = x2 u’ = 2x2 – 1 = 2x & v = 3x – 1 v’ = 3(1) – 0 = 3 f’2(x) = (𝑢/𝑣)^′ (xn)’ = nxn – 1 & (a)’ = 0 where a is constant = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 = (2𝑥(3𝑥 − 1) − 3 (𝑥2))/(3𝑥 − 1)2 = (6𝑥2 − 2𝑥 − 3𝑥2)/〖(3𝑥 − 1)〗^2 = (3𝑥2 − 2𝑥 )/〖(3𝑥 − 1)〗^2 = (𝑥(3𝑥 − 2))/〖(3𝑥 − 1)〗^2 Hence f’2(x) = (𝑥 (3𝑥 − 2))/(3𝑥 − 1)2 Now f’ (x) = f1’(x) – f2’ (x) = (−𝟐)/(𝒙 + 𝟏)𝟐 – (𝒙(𝟑𝒙 − 𝟐))/(𝟑𝒙 − 𝟏)𝟐