# NCERT solution class 12 chapter 5 Continuity and Differentiability exercise 5.9 mathematics part 1

## EXERCISE 5.9

#### Question 1:

Using chain rule, we obtain

#### Question 3:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

#### Question 4:

Using chain rule, we obtain

#### Question 6:

Therefore, equation (1) becomes

#### Question 7:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

#### Question 8:

for some constant a and b.

By using chain rule, we obtain

#### Question 9:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

#### Question 10:

, for some fixed and

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

s = aa

Since a is constant, aa is also a constant.

From (1), (2), (3), (4), and (5), we obtain

#### Question 11:

, for

Differentiating both sides with respect to x, we obtain

Differentiating with respect to x, we obtain

Also,

Differentiating both sides with respect to x, we obtain

Substituting the expressions of in equation (1), we obtain

Find, if

Find, if

#### Question 14:

If, for, −1 < x <1, prove that

It is given that,

Differentiating both sides with respect to x, we obtain

Hence, proved.

#### Question 15:

If, for some  prove that

is a constant independent of a and b.

It is given that,

Differentiating both sides with respect to x, we obtain

Hence, proved.

#### Question 16:

If  with  prove that

Then, equation (1) reduces to

⇒sina+y-ydydx=cos2a+y⇒dydx=cos2a+ysina

Hence, proved.

If and, find

#### Question 18:

If, show that exists for all real x, and find it.

It is known that,

Therefore, when x ≥ 0,

In this case,  and hence,

When x < 0,

In this case,  and hence,

Thus, forexists for all real x and is given by,

#### Question 19:

Using mathematical induction prove that for all positive integers n.

For n = 1,

∴P(n) is true for n = 1

Let P(k) is true for some positive integer k.

That is,

It has to be proved that P(k + 1) is also true.

Thus, P(k + 1) is true whenever P (k) is true.

Therefore, by the principle of mathematical induction, the statement P(n) is true for every positive integer n.

Hence, proved.

#### Question 20:

Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.

Differentiating both sides with respect to x, we obtain

#### Question 21:

Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer ?

y=x           -∞<x≤1    2-x         1≤x≤∞ It can be seen from the above graph that, the given function is continuos everywhere but not differentiable at exactly two points which are 0 and 1.

If, prove that

Thus,

If, show that