Differentiate x sin x w.r.t sin x .

Let u = x sin x and v = sin x Differentiate u w.r.t x by using quotient rule, we get ∴ d u d x = sin x · d d x x - x · d d x sin x ( sin x ) 2 = sin x - x cos x sin 2 x and d v d x = d d x sin x = cos x ∴ d u d v = d u d x d v d x = sin x - x cos x sin 2 x cos x = sin x - x cos x sin 2 x cos x = sin x - x cos x cos x sin 2 x cos x cos x Dividing both numerator and denominator by cos x , we get, = tan x - x sin 2 x Thus d u d v = tan x - x sin 2 x .

If y = tan - 1 x , find d 2 y d x 2 in terms of y alone.

We have, y = tan - 1 x Differentiating w.r.t x , d y d x = 1 1 + x 2 Again differentiating w.r.t x , Now d 2 y d x 2 = d d x 1 + x 2 - 1 = - 1 1 + x 2 - 2 · d d x 1 + x 2 = - 1 1 + x 2 2 · 2 x = - 2 tan y 1 + tan 2 y 2 ∵ y = tan - 1 x ⇒ tan y = x = - 2 tan y sec 2 y 2 = - 2 · sin y cos y · cos 4 y ∴ d 2 y d x 2 = - sin 2 y · cos 2 y [ ∵ sin 2 x = 2 sin x cos x ] .

Find d y d x , if y = x tan x + x 2 + 1 2 .

Given that y = x tan x + x 2 + 1 2 . . . . . . i Taking u = x tan x and v = x 2 + 1 2 ⇒ log u = tan x log x . . . . . . ( ii ) and v 2 = x 2 + 1 2 . . . . . . . . ( iii ) On differentiating Eq. ( ii ) , w.r.t. x , we get 1 u · d u d x = tan x · 1 x + log x · sec 2 x ⇒ d u d x = u tan x x + log x · sec 2 x = x tan x tan x x + log x · sec 2 x . . . . ( iv ) Also, differentiating Eq. ( iii ) w.r.t. x we get 2 v · d v d x = 1 2 ( 2 x ) ⇒ d v d x = 1 4 v · ( 2 x ) ⇒ d v d x = 1 4 x 2 + 1 2 · 2 x = x · 2 2 x 2 + 1 ⇒ d v d x = x 2 x 2 + 1 . . . . . . . ( v ) Now y = u + v ⇒ d y d x = d u d x + d v d x ∴ d y d x = x tan x tan x x + log x · sec 2 x + x 2 x 2 + 1 .

E M B I B E

NCERT Exemplar Mathematics - Class 12>Chapter 5 - Continuity and Differentiability>EXERCISE

NCERT Solutions for Chapter: Continuity and Differentiability, Exercise 1: EXERCISE

Author:NCERT

NCERT Mathematics Solutions for Exercise - NCERT Solutions for Chapter: Continuity and Differentiability, Exercise 1: EXERCISE

Attempt the practice questions on Chapter 5: Continuity and Differentiability, Exercise 1: EXERCISE with hints and solutions to strengthen your understanding. NCERT Exemplar Mathematics - Class 12 solutions are prepared by Experienced Embibe Experts.