# Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 9 - Differentiation [Latest edition]

## Chapter 9: Differentiation

Exercise 9.1Exercise 9.2Miscellaneous Exercise 9
Exercise 9.1 [Pages 187 - 188]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 9 Differentiation Exercise 9.1 [Pages 187 - 188]

Exercise 9.1 | Q 1. (a) | Page 187

Find the derivative of the following w. r. t. x by using method of first principle:

x2 + 3x – 1

Exercise 9.1 | Q 1. (b) | Page 187

Find the derivative of the following w. r. t. x by using method of first principle:

sin (3x)

Exercise 9.1 | Q 1. (c) | Page 187

Find the derivative of the following w. r. t. x by using method of first principle:

e2x+1

Exercise 9.1 | Q 1. (d) | Page 187

Find the derivative of the following w. r. t. x by using method of first principle:

3x

Exercise 9.1 | Q 1. (e) | Page 187

Find the derivative of the following w. r. t. x by using method of first principle:

log (2x + 5)

Exercise 9.1 | Q 1. (f) | Page 187

Find the derivative of the following w. r. t. x by using method of first principle:

tan (2x + 3)

Exercise 9.1 | Q 1. (g) | Page 187

Find the derivative of the following w. r. t. x by using method of first principle:

sec (5x − 2)

Exercise 9.1 | Q 1. (h) | Page 187

Find the derivative of the following w. r. t. x by using method of first principle:

x sqrt(x)

Exercise 9.1 | Q 2. (a) | Page 187

Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:

sqrt(2x + 5) at x = 2

Exercise 9.1 | Q 2. (b) | Page 187

Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:

tan x at x = pi/4

Exercise 9.1 | Q 2. (c) | Page 187

Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:

2^(3x + 1) at x = 2

Exercise 9.1 | Q 2. (d) | Page 187

Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:

log(2x + 1) at x = 2

Exercise 9.1 | Q 2. (e) | Page 187

Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:

"e"^(3x - 4) at x = 2

Exercise 9.1 | Q 2. (f) | Page 187

Find the derivative of the following w. r. t. x. at the point indicated against them by using method of first principle:

cos x at x = (5pi)/4

Exercise 9.1 | Q 3 | Page 188

Show that the function f is not differentiable at x = −3, where f(x) {:(=  x^2 + 2, "for"  x < - 3),(= 2 - 3x, "for"  x ≥ - 3):}

Exercise 9.1 | Q 4 | Page 188

Show that f(x) = x2 is continuous and differentiable at x = 0

Exercise 9.1 | Q 5. (i) | Page 188

Discuss the continuity and differentiability of f(x) = x |x| at x = 0

Exercise 9.1 | Q 5. (ii) | Page 188

Discuss the continuity and differentiability of f(x) = (2x + 3) |2x + 3| at x = - 3/2

Exercise 9.1 | Q 6 | Page 188

Discuss the continuity and differentiability of f(x) at x = 2

f(x) = [x] if x ∈ [0, 4). [where [*] is a greatest integer (floor) function]

Exercise 9.1 | Q 7 | Page 188

Test the continuity and differentiability of f(x) {:(= 3 x + 2, "if"  x > 2),(= 12 - x^2, "if"  x ≤ 2):}} at x = 2

Exercise 9.1 | Q 8 | Page 188

If f(x) {:(= sin x - cos x, "if"  x ≤ pi/2),(= 2x - pi + 1, "if"  x > pi /2):} Test the continuity and differentiability of f at x = π/2

Exercise 9.1 | Q 9 | Page 188

Examine the function

f(x) {:(= x^2 cos (1/x)",", "for"  x ≠ 0),(= 0",", "for"  x = 0):}

for continuity and differentiability at x = 0

Exercise 9.2 [Page 192]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 9 Differentiation Exercise 9.2 [Page 192]

Exercise 9.2 | Q I. (1) | Page 192

Differentiate the following w.r.t.x :

y = x^(4/3) + "e"^x - sinx

Exercise 9.2 | Q I. (2) | Page 192

Differentiate the following w.r.t.x :

y = sqrt(x) + tan x - x^3

Exercise 9.2 | Q I. (3) | Page 192

Differentiate the following w.r.t.x :

y = log x - "cosec"  x + 5^x - 3/(x^(3/2))

Exercise 9.2 | Q I. (4) | Page 192

Differentiate the following w.r.t.x :

y = x^(7/3) + 5x^(4/5) - 5/(x^(2/5))

Exercise 9.2 | Q I. (5) | Page 192

Differentiate the following w.r.t.x :

y = 7^x + x^7 - 2/3 xsqrt(x) - logx + 7^7

Exercise 9.2 | Q I. (6) | Page 192

Differentiate the following w.r.t.x :

y = 3 cotx - 5"e"^x + 3logx - 4/(x^(3/4))

Exercise 9.2 | Q II. (1) | Page 192

Differentiate the following w.r.t.x. :

y = x5 tan x

Exercise 9.2 | Q II. (2) | Page 192

Differentiate the following w.r.t.x. :

y = x3 log x

Exercise 9.2 | Q II. (3) | Page 192

Differentiate the following w.r.t.x. :

y = (x2 + 2)2 sin x

Exercise 9.2 | Q II. (4) | Page 192

Differentiate the following w.r.t.x. :

y = ex logx

Exercise 9.2 | Q II. (5) | Page 192

Differentiate the following w.r.t.x. :

y = x^(3/2)  "e"^xlogx

Exercise 9.2 | Q II. (6) | Page 192

Differentiate the following w.r.t.x. :

y = log ex3 log x3

Exercise 9.2 | Q III. (1) | Page 192

Differentiate the following w.r.t.x. :

y = x^2sqrt(x) + x^4logx

Exercise 9.2 | Q III. (2) | Page 192

Differentiate the following w.r.t.x. :

y = "e"^xsecx - x^(5/3) log x

Exercise 9.2 | Q III. (3) | Page 192

Differentiate the following w.r.t.x. :

y = x^4 + x sqrt(x) cos x - x^2"e"^x

Exercise 9.2 | Q III. (4) | Page 192

Differentiate the following w.r.t.x. :

y = (x3 – 2) tan x – x cos x + 7x. x7

Exercise 9.2 | Q III. (5) | Page 192

Differentiate the following w.r.t.x. :

y = sinx logx + "e"^x cos x - "e"^x sqrt(x)

Exercise 9.2 | Q IiI. (6) | Page 192

Differentiate the following w.r.t.x. :

y = "e"^x tanx + cos x log x - sqrt(x)  5^x

Exercise 9.2 | Q IV. (1) | Page 192

Differentiate the following w.r.t.x. :

y = (x^2 + 3)/(x^2 - 5)

Exercise 9.2 | Q IV. (2) | Page 192

Differentiate the following w.r.t.x. :

y = (sqrt(x) + 5)/(sqrt(x) - 5)

Exercise 9.2 | Q IV. (3) | Page 192

Differentiate the following w.r.t.x. :

y = (x"e"^x)/(x + "e"^x)

Exercise 9.2 | Q IV. (4) | Page 192

Differentiate the following w.r.t.x. :

y = (x log x)/(x + log x)

Exercise 9.2 | Q IV. (5) | Page 192

Differentiate the following w.r.t.x. :

y = (x^2 sin x)/(x + cos x)

Exercise 9.2 | Q IV. (6) | Page 192

Differentiate the following w.r.t.x. :

y = (5"e"^x - 4)/(3"e"^x - 2)

Exercise 9.2 | Q V. (1) | Page 192

If f(x) is a quadratic polynomial such that f(0) = 3, f'(2) = 2 and f'(3) = 12 then find f(x)

Exercise 9.2 | Q V. (2) | Page 192

If f(x) = a sin x – b cos x, "f'"(pi/4) = sqrt(2) and "f'"(pi/6) = 2, then find f(x)

Exercise 9.2 | Q VI. (1) | Page 192

Fill in the blanks:

y = ex .tan x

Differentiating w.r.t.x

("d"y)/("d"x) = "d"/("d"x)("e"^x tan x)

= square "d"/("d"x) tanx + tan x "d"/("d"x) square

= square  square + tan x  square

= "e"^x [square  + square]

Exercise 9.2 | Q VI. (2) | Page 192

Fill in the blanks:

y = sinx/(x^2 + 2)

Differentiating. w.r.t.x.

("d"y)/("d"x) = (square "d"/("d"x) (sin x) - sin x "d"/("dx) square)/(x^2 + 2)^2

= (square  square - sin x  square)/(x^2 + 2)^2

= (square - square)/(x^2 + 2)^2

Exercise 9.2 | Q VI. (3) | Page 192

Fill in the blanks:

y = (3x2 + 5) cos x

Differentiating w.r.t.x

("d"y)/("d"x) = "d"/("d"x) [(3x^2 + 5) cos x]

= (3x^2 + 5) "d"/("d"x) [square] + cos x  "d"/("d"x) [square]

= (3x^2 + 5) [square] + cos x  [square]

∴ (dx)/("d"y) = (3x^2 + 5) [square] + [square] cos x

Exercise 9.2 | Q VI. (4) | Page 192

Fill in the blank:

Differentiate tan x and sec x w.r.t.x. using the formulae for differentiation of "u"/"v" and 1/"v" respectively

Miscellaneous Exercise 9 [Pages 194 - 195]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 9 Differentiation Miscellaneous Exercise 9 [Pages 194 - 195]

Miscellaneous Exercise 9 | Q I. (1) | Page 194

Select the correct answer from the given alternative:

If y = (x - 4)/(sqrtx + 2), then ("d"y)/("d"x)

• 1/(x + 4)

• sqrt(x)/((sqrt(x +  2))^2

• 1/(2sqrt(x))

• x/((sqrt(x) +  2)^2

Miscellaneous Exercise 9 | Q I. (2) | Page 194

Select the correct answer from the given alternative:

If y = ("a"x + "b")/("c"x + "d"), then ("d"y)/("d"x) =

• ("ab" - "cd")/("c"x + "d")^2

• ("a"x - "c")/("c"x + "d")^2

• ("ac" - "bd")/("c"x + "d")^2

• ("ad" - "bc")/("c"x + "d")^2

Miscellaneous Exercise 9 | Q I. (3) | Page 194

Select the correct answer from the given alternative:

If y = (3x + 5)/(4x + 5), then ("d"y)/("d"x) =

• -15/(3x + 5)^2

• -15/(4x + 5)^2

• -5/(4x + 5)^2

• -13/(4x + 5)^2

Miscellaneous Exercise 9 | Q I. (4) | Page 194

Select the correct answer from the given alternative:

If y = (5sin x - 2)/(4sin x + 3), then ("d"y)/("d"x) =

• (7 cos x)/(4 sin x + 3)^2

• (23 cos x)/(4 sin x + 3)^2

• - (7 cos x)/(4 sin x + 3)^2

• -(15 cos x)/(4 sin x + 3)^2

Miscellaneous Exercise 9 | Q I. (5) | Page 194

Select the correct answer from the given alternative:

Suppose f(x) is the derivative of g(x) and g(x) is the derivative of h(x).

If h(x) = a sin x + b cos x + c then f(x) + h(x) =

• 0

• c

• – c

• − 2(a sin + b cos x)

Miscellaneous Exercise 9 | Q I. (6) | Page 194

Select the correct answer from the given alternative:

If f(x) {:(= 2x + 6, "for"  0 ≤ x ≤ 2),(= "a"x^2 + "b"x, "for"  2 < x ≤4):} is differentiable at x = 2 then the values of a and b are

• a = - 3/2, b = 3

• a = 3/2, b = 8

• a = 1/2, b = 8

• a = - 3/2, b = 8

Miscellaneous Exercise 9 | Q I. (7) | Page 195

Select the correct answer from the given alternative:

If f(x) {:( = x^2 + sin x + 1, "for"  x ≤ 0),(= x^2 - 2x + 1, "for"  x ≤ 0):} then

• f is continuous at x = 0, but not differentiable at x = 0

• f is neither continuous nor differentiable at x = 0

• f is not continuous at x = 0, but differentiable at x = 0

• f is both continuous and differentiable at x = 0

Miscellaneous Exercise 9 | Q I. (8) | Page 195

Select the correct answer from the given alternative:

If, f(x) = x^50/50 + x^49/49 + x^48/48 + .... +x^2/2 + x + 1, thef f'(1) =

• 48

• 49

• 50

• 51

Miscellaneous Exercise 9 [Page 195]

### Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board Chapter 9 Differentiation Miscellaneous Exercise 9 [Page 195]

Miscellaneous Exercise 9 | Q II. (1) | Page 195

Determine whether the following function is differentiable at x = 3 where,

f(x) {:( = x^2 + 2","  ,  "for"  x ≥ 3),(= 6x - 7"," ,  "for"  x < 3):}

Miscellaneous Exercise 9 | Q II. (2) | Page 195

Find the values of p and q that make function f(x) differentiable everywhere on R

f(x) {:( = 3 - x"," , "for"  x < 1),(= "p"x^2 + "q"x",", "for"  x ≥ 1):}

Miscellaneous Exercise 9 | Q II. (3) | Page 195

Determine the values of p and q that make the function f(x) differentiable on R where

f(x) {:( = "p"x^3",", "for"  x < 2),(= x^2 + "q"",", "for"  x ≥ 2):}

Miscellaneous Exercise 9 | Q II. (4) | Page 195

Determine all real values of p and q that ensure the function

f(x) {:( = "p"x + "q"",", "for"  x ≤ 1),(= tan ((pix)/4)",", "for"  1 < x < 2):} is differentiable at x = 1

Miscellaneous Exercise 9 | Q II. (5) | Page 195

Discuss whether the function f(x) = |x + 1| + |x  – 1| is differentiable ∀ x ∈ R

Miscellaneous Exercise 9 | Q II. (6) | Page 195

Test whether the function f(x) {:(= 2x - 3",", "for"  x ≥ 2),(= x - 1",", "for"  x < 2):} is differentiable at x = 2

Miscellaneous Exercise 9 | Q II. (7) | Page 195

Test whether the function f(x) {:(= x^2 + 1",", "for"  x ≥ 2),(= 2x + 1",", "for"  x < 2):} is differentiable at x = 2

Miscellaneous Exercise 9 | Q II. (8) | Page 195

Test whether the function f(x) {:(= 5x - 3x^2",", "for"  x ≥ 1),(= 3 - x",", "for"  x < 1):} is differentiable at x = 1

Miscellaneous Exercise 9 | Q II. (9) | Page 195

If f(2) = 4, f′(2) = 1 then find lim_(x -> 2) [(x"f"(2) - 2"f"(x))/(x - 2)]

Miscellaneous Exercise 9 | Q II. (10) | Page 195

If y = "e"^x/sqrt(x) find ("d"y)/("d"x) when x = 1

## Chapter 9: Differentiation

Exercise 9.1Exercise 9.2Miscellaneous Exercise 9

## Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 9 - Differentiation

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Concepts covered in Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 9 Differentiation are Definition of Derivative and Differentiability, Rules of Differentiation (Without Proof), Derivative of Algebraic Functions, Derivatives of Trigonometric Functions, Derivative of Logarithmic Functions, Derivatives of Exponential Functions, L' Hospital'S Theorem.

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