## EXERCISE 5.9

#### Page No 191:

#### Question 1:

#### Answer:

Using chain rule, we obtain

#### Question 2:

#### Answer:

#### Question 3:

#### Answer:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to *x*, we obtain

#### Question 4:

#### Answer:

Using chain rule, we obtain

#### Question 5:

#### Answer:

#### Question 6:

#### Answer:

Therefore, equation (1) becomes

#### Question 7:

#### Answer:

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*.

#### Answer:

By using chain rule, we obtain

#### Question 9:

#### Answer:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to *x*, we obtain

#### Question 10:

, for some fixed and

#### Answer:

Differentiating both sides with respect to *x*, we obtain

Differentiating both sides with respect to *x*, we obtain

*s* = *a*^{a}

Since *a* is constant, *a*^{a} is also a constant.

∴

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

#### Question 11:

, for

#### Answer:

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

#### Question 12:

Find, if

#### Answer:

#### Question 13:

Find, if

#### Answer:

#### Question 14:

If, for, −1 < *x* <1, prove that

#### Answer:

_{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*.

#### Answer:

It is given that,

Differentiating both sides with respect to *x*, we obtain

Hence, proved.

#### Page No 192:

#### Question 16:

If with prove that

#### Answer:

Then, equation (1) reduces to

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

Hence, proved.

#### Question 17:

If and, find

#### Answer:

#### Question 18:

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

#### Answer:

It is known that,

Therefore, when *x* ≥ 0,

In this case, and hence,

When *x* < 0,

In this case, and hence,

Thus, for, exists for all real *x* and is given by,

#### Question 19:

Using mathematical induction prove that for all positive integers *n*.

#### Answer:

_{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.

#### Answer:

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 ?

#### 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.

#### Question 22:

If, prove that

#### Answer:

Thus,

#### Question 23:

If, show that

#### Answer:

It is given that,