If x = t 2 and y = t 3 , find d 2 y d x 2 .

We have, y = t 3 ⇒ d y d t = 3 t 2 And, x = t 2 ⇒ d x d t = 2 t Then, d y d x = d y d t d x d t ⇒ d y d x = 3 t 2 2 t = 3 2 t and, d 2 y d x 2 = 3 2 · d t d x ⇒ d 2 y d x 2 = 3 2 · 1 2 t = 3 4 t

If x = 2 a t , y = a t 2 where a is a constant, then find d 2 y d x 2 at x = 1 2 .

Here x = 2 a t and y = a t 2 Differentiating w.r.t t , we get d x d t = 2 a and d y d t = 2 a t ∴ d y d x = d y d t · d t d x = 2 a t 2 a = t Differentiating again w.r.t t , we get d 2 y d x 2 = d d x d y d x = d d x t = d d t t × d t d x = 1 2 a Now d 2 y d x 2 x = 1 2 = 1 2 a

If y = x + e x , find d 2 x d y 2 .

We have, y = x + e x Differentiating w.r.t x both sides, we get d y d x = d d x x + d d x e x ⇒ d y d x = 1 + e x ⇒ d x d y = 1 1 + e x Again, differentiating w.r.t y both sides, we get ⇒ d 2 x d y 2 = - e x 1 + e x 2 · d x d y = - e x 1 + e x 3

E M B I B E

MATHEMATICS CLASS XII VOLUME-1>Chapter 12 - Higher Order Derivatives>MULTIPLE CHOICE QUESTIONS (MCQs)

R D Sharma Solutions for Chapter: Higher Order Derivatives, Exercise 1: MULTIPLE CHOICE QUESTIONS (MCQs)

Author:R D Sharma

R D Sharma Mathematics Solutions for Exercise - R D Sharma Solutions for Chapter: Higher Order Derivatives, Exercise 1: MULTIPLE CHOICE QUESTIONS (MCQs)

Attempt the free practice questions on Chapter 12: Higher Order Derivatives, Exercise 1: MULTIPLE CHOICE QUESTIONS (MCQs) with hints and solutions to strengthen your understanding. MATHEMATICS CLASS XII VOLUME-1 solutions are prepared by Experienced Embibe Experts.