## NCERT Solutions for Class 12 Maths Chapter 5 – Continuity and Differentiability Ex 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_{}

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