Find the derivative of the function 1 - x x 1 + x x x > 0 .

Given, d d x 1 - x 3 2 x 3 2 + 1 = d d x 1 - x 3 2 · x 3 2 + 1 - 1 - x 3 2 · d d x x 3 2 + 1 x 3 2 + 1 2 = 0 - 3 2 x 3 2 - 1 x 3 2 + 1 - 1 - x 3 2 3 2 x 3 2 - 1 + 0 x 3 2 + 1 2 = - 3 x x 3 2 + 1 2 - 3 x 1 - x 3 2 2 x 3 2 + 1 2 ∴ d d x 1 - x x 1 + x x = - 3 x x 3 2 + 1 2

Find the derivative of the function 1 x - x 3 e x .

Given, d d x 1 x - x 3 e x = d d x 1 x - x 3 · e x + 1 x - x 3 · d d x e x = 3 1 x - x 2 · d d x 1 x - x · e x + 1 x - x 3 e = 3 1 x - x 2 - 1 x 2 - 1 e x + 1 x - x 3 e x ∴ d dx 1 x - x 3 e x = 3 1 x - x 2 - 1 x 2 - 1 e x + 1 x - x 3 e x

Show that the function f x = | x | + | x - 1 | , x ∈ R is differentiable for all real number except for 0 and 1 .

Given, f ( x ) = | x | + | x + 1 | , ∀ x ∈ R Case (i): At x = 0 R f ' 0 = lim x → 0 + f ( x ) - f ( 0 ) x = lim x → 0 + 2 x + 1 - 1 x = 2 L f ' 0 lim x → 0 - f ( x ) - f ( 0 ) x = lim x → 0 - 1 - 1 x = 0 R f ' ( 0 ) ≠ L f ' ( 0 ) ∴ f ' 0 does not exist. f x is not differentiable at x = 0 Case (ii): At x = 1 R f ' 1 = lim x → 1 + f ( x ) - f ( 1 ) x - 1 = lim x → 1 + 2 x + 1 - 3 x - 1 = lim x → 1 + 2 x - 2 x - 1 = 2 L f ' 1 = lim x → 1 - f ( x ) - f ( 1 ) x - 1 = lim x → 1 - 1 - 1 x - 1 = 0 R f ' ( 1 ) ≠ L f ' ( 1 ) f ' ( 1 ) does not exist. f ( x ) is not differentiable at x = 1 ∴ f ( x ) is differentiable on R - { 0 , 1 } .

Verify whether the function f x = x if x < 1 3 - x if 1 ≤ x ≤ 3 x 2 - 4 x + 3 if x > 3 is differentiable at 1 and 3 .

Given, f x = x if x < 1 3 - x if 1 ≤ x ≤ 3 x 2 - 4 x + 3 if x > 3 A function f x is differentiable at x = a , if it is continuous at x = a and LHD = RHD at x = a . LHD and RHD of f x at x = a are given by LHD = lim h → 0 - f ( a - h ) - f ( a ) - h and RHD = lim h → 0 + f ( a + h ) - f ( a ) h . Case (i): At x = 1 f ' 1 - = lim x → 1 - f ( x ) - f ( 1 ) x - 1 = lim x → 1 - x - ( 3 - 1 ) x - 1 = lim x → 1 - x - 2 x - 1 = ∞ f ' 1 + = lim x → 1 + f ( x ) - f ( 1 ) x - 1 = lim x → 1 + ( 3 - x ) - 2 x - 1 = lim x → 1 + 1 - x x - 1 = - 1 f ' 1 - ≠ f ' 1 + f x is not differentiable at x = 1 Case (ii): At x = 3 f ' 3 - = lim x → 3 - f ( x ) - f ( 3 ) x - 3 = lim x → 3 - ( 3 - x ) - 0 x - 3 = lim x → 3 - 3 - x x - 3 = - 1 f ' 3 + = lim x → 3 + f ( x ) - f ( 3 ) x - 3 = lim x → 3 + x 2 - 4 x + 3 - 0 x - 3 = lim x → 3 + ( x - 3 ) ( x - 1 ) x - 3 = lim x → 3 + x - 1 = 3 - 1 = 2 f ' 3 - ≠ f ' 3 + f x is not differentiable at x = 3 .

Is the function f x = x if 0 ≤ x ≤ 2 2 if x ≥ 2 derivable at 2 ? Justify.

Given, f x = x , if 0 ≤ x ≤ 2 2 , if x ≥ 2 A function f x is differentiable at x = a , if it is continuous at x = a and LHD = RHD at x = a . LHD and RHD of f x at x = a are given by LHD = lim h → 0 - f ( a - h ) - f ( a ) - h and RHD = lim h → 0 + f ( a + h ) - f ( a ) h . f ' 2 - = lim x → 2 - f ( x ) - f ( 2 ) x - 2 = lim x → 2 - x - 2 x - 2 = 1 f ' 2 + = lim x → 2 + f ( x ) - f ( 2 ) x - 2 = lim x → 2 + 2 - 2 x - 2 = 0 f ' 2 - ≠ f ' 2 + f x is not derivable at x = 2 .

E M B I B E

Intermediate First Year Mathematics Paper 1B>Chapter 9 - Differentiation>Exercise 9(a)

Andhra Pradesh Board Solutions for Chapter: Differentiation, Exercise 1: Exercise 9(a)

Author:Andhra Pradesh Board

Andhra Pradesh Board Mathematics Solutions for Exercise - Andhra Pradesh Board Solutions for Chapter: Differentiation, Exercise 1: Exercise 9(a)

Attempt the practice questions on Chapter 9: Differentiation, Exercise 1: Exercise 9(a) with hints and solutions to strengthen your understanding. Intermediate First Year Mathematics Paper 1B solutions are prepared by Experienced Embibe Experts.

Questions from Andhra Pradesh Board Solutions for Chapter: Differentiation, Exercise 1: Exercise 9(a) with Hints & Solutions

MEDIUM

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 9 - Differentiation>Exercise 9(a)>Q II

Find the derivative of the function $f (x) = \cos^{2} x$ from the first principle.

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 9 - Differentiation>Exercise 9(a)>Q II

Find the derivative of the function $\frac{1 - x \sqrt{x}}{1 + x \sqrt{x}} (x > 0)$ .

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 9 - Differentiation>Exercise 9(a)>Q II

Find the derivative of the function $x^{n} n^{x} \log (n x) (x > 0, n \in N)$ .

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 9 - Differentiation>Exercise 9(a)>Q II

Find the derivative of the function $a x^{2 n} \log x + b x^{n} e^{- x}$ .

EASY

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 9 - Differentiation>Exercise 9(a)>Q II

Find the derivative of the function ${(\frac{1}{x} - x)}^{3} e^{x}$ .

MEDIUM

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 9 - Differentiation>Exercise 9(a)>Q II

Show that the function $f (x) = | x | + | x - 1 |, x \in R$ is differentiable for all real number except for $0$ and $1$ .

MEDIUM

11th Andhra Pradesh Board

IMPORTANT

Intermediate First Year Mathematics Paper 1B>Chapter 9 - Differentiation>Exercise 9(a)>Q II

Verify whether the function $f (x) = \{\begin{matrix} x & if & x < 1 \\ 3 - x & if & 1 \leq x \leq 3 \\ x^{2} - 4 x + 3 & if & x > 3 \end{matrix}$ is differentiable at $1$ and $3$ .