Balbharti Maharashtra State Board 11th Maths Book Solutions Pdf Chapter 9 Differentiation Ex 9.1 Questions and Answers.

## Maharashtra State Board 11th Maths Solutions Chapter 9 Differentiation Ex 9.1

Question 1.

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

(a) x^{2} + 3x – 1

Solution:

Let f(x) = x^{2} + 3x – 1

∴ f(x + h) = (x + h)^{2} + 3(x + h) – 1

= x^{2} + 2xh + h^{2} + 3x + 3h – 1

By first principle, we get

(b) sin(3x)

Solution:

Let f(x) = sin 3x

f(x + h) = sin3(x + h) = sin(3x + 3h)

By first principle, we get

(c) e^{2x+1}

Solution:

(d) 3^{x}

Solution:

(e) log(2x + 5)

Solution:

Let f(x) = log(2x + 5)

∴ f(x + h) = log[2(x + h) + 5] = log (2x + 2h + 5)

By first principle, we get

(f) tan(2x + 3)

Solution:

Let f(x) = tan(2x + 3)

∴ f(x + h) = tan[2(x + h) + 3] = tan(2x + 2h + 3)

By first principle, we get

(g) sec(5x – 2)

Solution:

Let f(x) = sec(5x – 2)

f(x + h) = sec[5(x + h) – 2] = sec(5x + 5h – 2)

By first principle, we get

(h) x√x

Solution:

Question 2.

Find the derivatives of the following w.r.t. x. at the points indicated against them by using the method of the first principle.

(i) \(\sqrt{2 x+5}\) at x = 2

Solution:

(ii) tan x at x = \(\frac{\pi}{4}\)

Solution:

(iii) 2^{3x+1} at x = 2

Solution:

(iv) log(2x + 1) at x = 2

Solution:

Let f(x) = log(2x + 1)

∴ f(2) = log [2(2) + 1] = log 5 and

f(2 + h) = log [2(2 + h) + 1] = log(2h + 5)

By first principle, we get

(v) e^{3x-4} at x = 2

Solution:

(vi) cos x at x = \(\frac{5 \pi}{4}\)

Solution:

Question 3.

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

Solution:

∴ L f'(-3) ≠ R f'(-3)

∴ f is not differentiable at x = -3.

Question 4.

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

Solution:

Question 5.

Discuss the continuity and differentiability of

(i) f(x) = x |x| at x = 0

Solution:

(ii) f(x) = (2x + 3) |2x + 3| at x = \(-\frac{3}{2}\)

Solution:

Question 6.

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]

Solution:

Explanation:

x ∈ [0, 4)

∴ 0 ≤ x < 4

We will plot graph for 0 ≤ x < 4

not for x < 0 and upto x = 4 making on X-axis.

f(x) = [x]

∴ Greatest integer function is discontinuous at all integer values of x and hence not differentiable at all integers.

∴ f is not continuous at x = 2.

∵ f(x) = 1, x < 2

= 2, x ≥ 2

x ∈ neighbourhood of x = 2.

∴ L.H.L. = 1, R.H.L. = 2

∴ f is not continuous at x = 2.

∴ f is not differentiable at x = 2.

Question 7.

Test the continuity and differentiability of

f(x) = 3x + 2 if x > 2

= 12 – x^{2} if x ≤ 2 at x = 2.

Solution:

Question 8.

If f(x) = sin x – cos x if x ≤ \(\frac{\pi}{2}\)

= 2x – π + 1 if x > \(\frac{\pi}{2}\)

Test the continuity and differentiability of f at x = \(\frac{\pi}{2}\).

Solution:

Question 9.

Examine the function

f(x) = x^{2} cos(\(\frac{1}{x}\)), for x ≠ 0

= 0, for x = 0

for continuity and differentiability at x = 0.

Solution: