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)

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

Author:Telangana Board

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

Attempt the free 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 Telangana Board Solutions for Chapter: Differentiation, Exercise 1: Exercise 9(a) with Hints & Solutions

MEDIUM

11th Telangana 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 Telangana 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 Telangana 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 Telangana 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 Telangana 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 Telangana 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 Telangana 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 $3$ .