#### Chapters

## Chapter 9: Differentiation

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

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

x^{2} + 3x – 1

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

sin (3x)

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

e^{2x+1}

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

3^{x}

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

log (2x + 5)

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

tan (2x + 3)

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

sec (5x − 2)

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

`x sqrt(x)`

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

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`

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

log(2x + 1) at x = 2

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

cos x at x = `(5pi)/4`

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):}`

Show that f(x) = x^{2} is continuous and differentiable at x = 0

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

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

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]

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

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`

Examine the function

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

for continuity and differentiability at x = 0

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

Differentiate the following w.r.t.x :

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

Differentiate the following w.r.t.x :

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

Differentiate the following w.r.t.x :

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

Differentiate the following w.r.t.x :

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

Differentiate the following w.r.t.x :

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

Differentiate the following w.r.t.x :

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

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

y = x^{5} tan x

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

y = x^{3} log x

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

y = (x^{2} + 2)^{2} sin x

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

y = e^{x} logx

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

y = `x^(3/2) "e"^xlogx`

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

y = log e^{x3} log x^{3}

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

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

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

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

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

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

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

y = (x^{3} – 2) tan x – x cos x + 7^{x}. x^{7}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Fill in the blanks:

y = e^{x} .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]`

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`

Fill in the blanks:

y = (3x^{2} + 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`

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

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

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`

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`

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`

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`

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)

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

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

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

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

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):}`

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):}`

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

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

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

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

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

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

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

## Chapter 9: Differentiation

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

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board chapter 9 (Differentiation) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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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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