**(I) Differentiate the following w.r.t. x**

**Question 1.**

Solution:

**Question 2.y = √x + tan x – x ^{3}Solution:**

**Question 3.Solution:**

**Question 4.Solution:**

**Question 5.Solution:**

**Question 6.Solution:**

**(II) Diffrentiate the following w.r.t. x**

**Question 1.y = x ^{5} tan xSolution:**

**Question 2.y = x ^{5} log xSolution:**

**Question 3.y = (x ^{2} + 2)^{2} sin xSolution:**

**Question 4.y = e ^{x} log xSolution:**

**Question 5.Solution:**

**Question 6.Solution:**

**(III) Diffrentiate the following w.r.t. x**

**Question 1.y = x ^{2}√x + x^{4} log xSolution:**

**Question 2.Solution:**

**Question 3.y = x ^{4} + x√x cos x – x^{2} e^{x}Solution:**

**Question 4.y = (x ^{3} – 2) tan x – x cos x + 7^{x} . x^{7}Solution:**

**Question 5.y = sin x log x + e ^{x} cos x – e^{x} √xSolution:**

**Question 6.y = e ^{x} tan x + cos x log x – √x 5^{x}Solution:**

**(IV) Differentiate the following w.r.t.x.**

**Question 1.Solution:**

**Question 2.Solution:**

**Question 3.Solution:**

**Question 4.**

**Question 5.Solution:**

**Question 6.Solution:**

**(V).**

**Question 1.If f(x) is a quadratic polynomial such that f(0) = 3, f'(2) = 2 and f'(3) = 12, then find f(x).Solution:**

Let f(x) = ax

^{2}+ bx + c …..(i)

∴ f(0) = a(0)

^{2}+ b(0) + c

∴ f(0) = c

But, f(0) = 3 …..(given)

∴ c = 3 …..(ii)

Differentiating (i) w.r.t. x, we get

f'(x) = 2ax + b

∴ f'(2) = 2a(2) + b

∴ f'(2) = 4a + b

But, f'(2) = 2 …..(given)

∴ 4a + b = 2 …..(iii)

Also, f'(3) = 2a(3) + b

∴ f'(3) = 6a + b

But, f'(3) = 12 …..(given)

∴ 6a + b = 12 …..(iv)

equation (iv) – equation (iii), we get

2a = 10

∴ a = 5

Substituting a = 5 in (iii), we get

4(5) + b = 2

∴ b = -18

∴ a = 5, b = -18, c = 3

∴ f(x) = 5x

^{2}– 18x + 3

Check:

If f(0) = 3, f'(2) = 2 and f'(3) = 12, then our answer is correct.

f(x) = 5x^{2} – 18x + 3 and f'(x) = 10x – 18

f(0) = 5(0)^{2} – 18(0) + 3 = 3

f'(2) = 10(2) – 18 = 2

f'(3) = 10(3) – 18 = 12

Thus, our answer is correct.

**Question 2.Solution:**

f(x) = a sin x – b cos x

Differentiating w.r.t. x, we get

f'(x) = a cos x – b (- sin x)

∴ f'(x) = a cos x + b sin x

Now, f(x) = a sin x – b cos x

∴ f(x) = (√3 + 1) sin x + (√3 – 1) cos x

**VI. Fill in the blanks. (Activity Problems)**

**Question 1.y = e ^{x} . tan xDiff. w.r.t. xSolution:**

**Question 2.diff. w.r.t. xSolution:**

**Question 3.y = (3x ^{2} + 5) cos xDiff. w.r.t. xSolution:**

**Question 4.Differentiate tan x and sec x w.r.t. x using the formulae for differentiation of and respectively.Solution:**