# Chapter 5 Miscellaneous (differentiation )

**Differentiate w.r.t. x the function in Exercises 1 to 11.(Class 12 ncert solution math chapter 5 miscellaneous)**

**Question 1:**

**Solution:** Let

Differentiate with respect to x

**Question 2:**

**Solution:** Let

Differentiate with respect to x

**Question 3:**

**Solution:** Let

Taking ‘log’ both side

**Question 4:**

**Solution:** Let

Differentiate with respect to x

**Question 5:**

**Solution :** Let

Differentiate with respect to x

**Question6:**

**Solution:** Let

Then,

Therefore from equation (i)

Differentiate with respect to x

**Question 7:**

**Solution:** Let

Taking log both side, we obtain

Differentiating both sides with respect to , we obtain

**Question 8:** , for some constant and .

**Solution:** Let

Using chain rule, we get

**Question 9:**

**Solution:** Let

Taking log on both the sides, we obtain

Differentiating both sides with respect to , we obtain

**Question 10:** Differentiate with respect to the function , for some fixed and .

**Solution:** Let

Also, let and

Therefore,

Now,

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Now,

Hence,

Now,

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Now,

Since is constant, is also a constant.

Hence,

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

**Question 11:** , for .

**Solution:** Let

Also, let and Therefore,

Now, Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

Now,

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to , we obtain

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

**Question 12:** Find , if

**Solution: **The given function is

Hence,

Therefore,

**Question 13:** Find , if .

**Solution:** The given function is

Let

Let

Hence,

From (i)

Differentiate with respect to x

**Question 14:** If for , prove that .

**Solution:**The given function is

Squaring both sides, we obtain

Differentiating both sides with respect to , we obtain

Hence proved.

**Question 15:**If for , prove that

is a constant independent of and .

**Solution:**The given function is

Differentiating both sides with respect to , we obtain

Therefore,

Hence,

is a constant and is independent of and .

Hence proved.

**Question 16:** If with , prove that .

**Solution: **The given function is

Therefore,

Since,

Then, equation (1) becomes,

Hence proved.

**Question 17:** If and , find .

**Solution:** The given function is and Therefore,

**Question 18:** If , show that exists for all real , and find it.

**Solution:** It is known that

Therefore, when

In this case, and hence,

When

In this case, and hence,

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

**Question 19:** Using mathematical induction prove that for all positive integers .

**Solution:**To prove: for all positive integers .

For ,

Therefore, is true for .

Let is true for some positive integer .

That is,

It has to be proved that is also true.

Taking, L.H.S.

Thus, is true whenever is true.

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

Hence, proved.

**Question 20:** Using the fact that and the differentiation, obtain the sum formula for cosines.

**Solution:** Given,

Differentiating both sides with respect to , we obtain

**Question 21:** Does there exist a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer?

**Solution:** Consider,

It can be seen from the above graph that the given function is continuous everywhere but not differentiable at exactly two points which are 0 and 1.

**Question 22:**

**Solution:**

Differentiate with respect to x

Hence proved

**Question 23:** If Show that

**Solution:**

Differentiate with respect to x

Multiply by both side

Squaring both side

Again differentiate with respect to x

Divide by both side