**Chapter 1 Differentiation Ex 1.1**

**Chapter 1 Differentiation Ex 1.1**

**Question 1.Differentiate the following w.r.t. x :(i) Solution:**

Method 1:

Let y =

Put u = – 2x – 1. Then y =

**Method 2:Let y = **

Differentiating w.r.t. x, we get

**Solution:**

Differentiating w.r.t. x, we get

**Question 2.**

Differentiating w.r.t. x, we get

Differentiating w.r.t. x, we get

**(v) **

**Solution:Let y = **

Differentiating w.r.t. x, we get

**(vi) Solution:Let y = **

Differentiating w.r.t. x, we get

Differentiating w.r.t. x, we get

**(viii) log[cos ( – 5)]Solution:Let y = log[cos ( – 5)]**

Differentiating w.r.t. x, we get

**(ix) Solution:Let y = **

Differentiating w.r.t. x, we get

**(x) [log (+ 7)]Solution:Let y = [log (+ 7)]**

Differentiating w.r.t. x, we get

**(xi) tan[cos (sinx)]Solution:Let y = tan[cos (sinx)]**

Differentiating w.r.t. x, we get

**(xii) sec[tan ( + 4)]Solution:Let y = sec[tan ( + 4)]**

Differentiating w.r.t. x, we get

Differentiating w.r.t. x, we get

**(xv) log[sec()]Solution:Let y = log[sec()]**

Differentiating w.r.t. x, we get

**(xvii) [log{log(logx)}Solution:let y = [log{log(logx)}**
Differentiating w.r.t. x, we get

**(xviii) Solution:Let y = **

Differentiating w.r.t. x, we get

**Question 3.Diffrentiate the following w.r.t. x(i) (x**

^{2}+ 4x + 1 Solution:

Let y = (x

^{2}+ 4x + 1

Differentiating w.r.t. x, we get

**(v) (1 + **

**Solution:**

Let y = (1 +

Differentiating w.r.t. x, we get

**(vii) log(sec 3x+ tan 3x)Solution:**

Let y = log(sec 3x+ tan 3x)

Differentiating w.r.t. x, we get

Solution:

Differentiating w.r.t. x, we get

Solution:

Differentiating w.r.t. x, we get

Differentiating w.r.t. x, we get

= 6cosec2x + 4 cotx +

Solution:

log = b log a

**Solution:**

Solution:

Solution:

**Solution:**

Differentiating w.r.t. x, we get

**Question 4.A table of values of f, g, f ‘ and g’ is given(i) If r(x) = f [g(x)] find r’ (2).Solution:**

r(x) = f[g(x)]

= f'[g(x)∙[g'(x)]

∴ r'(2) = f'[g(2)]∙g'(2)

= f'(6)∙g'(2) … [∵ g(x) = 6, when x = 2]

= -4 × 4 … [From the table]

= -16.

**(ii) If R(x) = g[3 + f(x)] find R’ (4).Solution:**

R(x) = g[3 + f(x)]

= g'[3 +f(x)]∙[0 + f'(x)]

= g'[3 + f(x)]∙f'(x)

∴ R'(4) = g'[3 + f(4)]∙f'(4)

= g'[3 + 3]∙f'(4) … [∵ f(x) = 3, when x = 4]

= g'(6)∙f'(4)

= 7 × 5 … [From the table]

= 35.

**(iii) If s(x) = f[9− f(x)] find s’ (4).Solution:**

s(x) = f[9− f(x)]

= f'[9 – f(x)]∙[0 – f'(x)]

= -f'[9 – f(x)] – f'(x)

∴ s'(4) = -f'[9 – f(4)] – f'(4)

= -f'[9 – 3] – f'(4) … [∵ f(x) = 3, when x = 4]

= -f'(6) – f'(4)

= -(-4)(5) … [From the table]

= 20.

**(iv) If S(x) = g[g(x)] find S’ (6)Solution:**

S(x) = g[g(x)]

= g'[g(x)]∙g'(x)

∴ S ‘(6) = g'[g'(6)]∙g'(6)

= g'(2)∙g'(6) … [∵ g (x) = 2, when x = 6]

= 4 × 7 … [From the table]

= 28.

**Question 5.**

**Question 6.Solution:**

Given f(1) = 4, g(1) = 3, f ‘(1) = 3, g'(1) = 4 …..(1)

**Question 7.Find the x co-ordinates of all the points on the curve y = sin 2x – 2 sin x, 0 ≤ x < 2π where dy/dx= 0.Solution:**

y = sin 2x – 2 sin x, 0 ≤ x < 2π