If y = x x , find d y d x at x = e ?

We have, y = x x ........(i) Taking log on both sides, log y = log x x ⇒ log y = x log x Differentiating both side w.r.t x ⇒ 1 y d y d x = x d d x log x + log x d d x x ⇒ 1 y d y d x = x 1 x + log x 1 ⇒ 1 y d y d x = 1 + log x ⇒ d y d x = y 1 + log x ⇒ d y d x = x x 1 + log x ........[Using equation (i)] Putting x = e , we get d y d x = e e 1 + log e e ⇒ d y d x = e e 1 + 1 . . . . . . . . . . . . . ∵ log e e = 1 ⇒ d y d x = 2 e e

If y = tan - 1 1 - x 1 + x , find d y d x .

Given y = tan - 1 1 - x 1 + x Differentiating both side w.r.t x using chain rule in differentiation d y d x = d d x tan - 1 1 - x 1 + x ⇒ d y d x = 1 1 + 1 - x 1 + x 2 d d x 1 - x 1 + x Now using quotient rule in differentiation ⇒ d y d x = 1 + x 2 1 + x 2 + 1 - x 2 1 + x d d x 1 - x - 1 - x d d x 1 + x 1 + x 2 ⇒ d y d x = 1 + x 2 2 x 2 + 2 1 + x - 1 - 1 - x 1 1 + x 2 ⇒ d y d x = 1 + x 2 2 x 2 + 2 - 2 1 + x 2 ⇒ d y d x = - 2 2 x 2 + 2 ⇒ d y d x = - 1 x 2 + 1

If y = log a x , find d y d x ?

We have, y = log a x ⇒ y = log x log a ∵ log a b = log b log a Differentiating both side w.r.t x we get, ⇒ d y d x = 1 log a d d x log x ⇒ d y d x = 1 log a 1 x ⇒ d y d x = 1 x log a

If y = log tan x write d y d x

We have, y = log tan x ⇒ y = log tan x 1 2 ⇒ y = 1 2 log tan x ∵ log a b = b log a Differentiating both side w.r.t x we get ⇒ d y d x = 1 2 × 1 tan x d d x tan x ⇒ d y d x = 1 2 × 1 tan x sec 2 x ⇒ d y d x = 1 2 sin x cos x × cos 2 x ⇒ d y d x = 1 2 sin x cos x ⇒ d y d x = 1 sin 2 x ⇒ d y d x = cosec 2 x

E M B I B E

MATHEMATICS CLASS XII VOLUME-1>Chapter 11 - Differentiation>MULTIPLE CHOICE QUESTIONS (MCQs)

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

Author:R D Sharma

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

Attempt the free practice questions on Chapter 11: Differentiation, 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.

Questions from R D Sharma Solutions for Chapter: Differentiation, Exercise 1: MULTIPLE CHOICE QUESTIONS (MCQs) with Hints & Solutions

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 11 - Differentiation>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 16

If $y = x^{x}$ , find $\frac{d y}{d x}$ at $x = e$ ?

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 11 - Differentiation>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 17

If $y = \tan^{- 1} (\frac{1 - x}{1 + x})$ , find $\frac{d y}{d x}$ .

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 11 - Differentiation>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 18

If $y = \log_{a} x$ , find $\frac{d y}{d x}$ ?

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 11 - Differentiation>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 19

If $y = \log \sqrt{\tan x}$ write $\frac{d y}{d x}$

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 11 - Differentiation>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 20

If $y = \sin^{- 1} (\frac{1 - x^{2}}{1 + x^{2}}) + \cos^{- 1} (\frac{1 - x^{2}}{1 + x^{2}})$ , find the value of $\frac{d y}{d x}$

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 11 - Differentiation>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 21

If $y = \sec^{- 1} (\frac{x + 1}{x - 1}) + \sin^{- 1} (\frac{x - 1}{x + 1})$ then write the value of $\frac{d y}{d x}$

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 11 - Differentiation>VERY SHORT ANSWER QUESTIONS (VSAQs)>Q 23

If $u = \sin^{- 1} (\frac{2 x}{1 + x^{2}})$ and $v = \tan^{- 1} (\frac{2 x}{1 - x^{2}})$ where $- 1 < x < 1$ , then write the value of $\frac{d u}{d v}$