## EXERCISE 5.2

#### Page No 166:

#### Question 1:

Differentiate the functions with respect to *x*.

#### Answer:

Let f(x)=sinx2+5, ux=x2+5, and v(t)=sint

Then, vou=vux=vx2+5=tanx2+5=f(x)

Thus, *f* is a composite of two functions.

**Alternate method**

#### Question 2:

Differentiate the functions with respect to *x*.

#### Answer:

Thus, *f *is a composite function of two functions.

Put *t* = *u* (*x*) = sin *x*

By chain rule,

**Alternate method**

#### Question 3:

Differentiate the functions with respect to *x*.

#### Answer:

Thus, *f *is a composite function of two functions, *u* and *v*.

Put *t* = *u* (*x*) = *ax* + *b*

Hence, by chain rule, we obtain

**Alternate method**

#### Question 4:

Differentiate the functions with respect to *x*.

#### Answer:

Thus, *f *is a composite function of three functions, *u, v*, and *w*.

Hence, by chain rule, we obtain

**Alternate method**

#### Question 5:

Differentiate the functions with respect to *x*.

#### Answer:

The given function is, where *g* (*x*) = sin (*ax* + *b*) and

*h* (*x*) = cos (*cx *+ *d*)

∴ *g *is a composite function of two functions, *u* and *v*.

Therefore, by chain rule, we obtain

∴*h* is a composite function of two functions, *p* and *q*.

Put *y* = *p* (*x*) = *cx *+ *d*

Therefore, by chain rule, we obtain

#### Question 6:

Differentiate the functions with respect to *x*.

#### Answer:

The given function is.

#### Question 7:

Differentiate the functions with respect to *x*.

#### Answer:

#### Question 8:

Differentiate the functions with respect to *x*.

#### Answer:

Clearly, *f *is a composite function of two functions, *u *and* v*, such that

By using chain rule, we obtain

**Alternate method**

#### Question 9:

Prove that the function *f *given by

is notdifferentiable at *x* = 1.

#### Answer:

The given function is

It is known that a function *f* is differentiable at a point *x* = *c* in its domain if both

are finite and equal.

To check the differentiability of the given function at *x* = 1,

consider the left hand limit of *f* at *x* = 1

Since the left and right hand limits of *f* at *x* = 1 are not equal, *f* is not differentiable at *x* = 1

#### Question 10:

Prove that the greatest integer function defined byis not

differentiable at *x* = 1 and *x* = 2.

#### Answer:

The given function *f* is

It is known that a function *f* is differentiable at a point *x* = *c* in its domain if both

are finite and equal.

To check the differentiability of the given function at *x* = 1, consider the left hand limit of *f* at *x* = 1

Since the left and right hand limits of *f* at *x* = 1 are not equal, *f* is not differentiable at

*x* = 1

To check the differentiability of the given function at *x* = 2, consider the left hand limit

of *f* at *x* = 2

Since the left and right hand limits of *f* at *x* = 2 are not equal, *f* is not differentiable at *x* = 2