NCERT Solutions Exercise 5.5: Continuity & Differentiability - Mathematics (Maths) for JEE Main & Advanced

Continuity & Differentiability 

Question 1: Differentiate the function with respect to x.

Answer

Question 2: Differentiate the function with respect to x.

Answer

Let y =
Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain 

Question 3: Differentiate the function with respect to x. 

Answer

Let y =
Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Question 4: Differentiate the function with respect to x. x^x - 2sinx

Answer

Let y =

u = x^x
Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Question 5: Differentiate the function with respect to x.

Answer

Let y =
Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Question 6: Differentiate the function with respect to x.

Answer

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

Therefore, from (1), (2), and (3), we obtain

Question 7: Differentiate the function with respect to x.

Answer

Let y =

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

Question 8: Differentiate the function with respect to x.

Answer

Let y =

Differentiating both sides with respect to x, we obtain

Therefore, from (1), (2), and (3), we obtain

Question 9: Differentiate the function with respect to x.

Answer
Let y =

Differentiating both sides with respect to x, we obtain


Question 10: Differentiate the function with respect to x.

Answer

Let y =

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

Question 11: Differentiate the function with respect to x.

Answer

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

 

Question 12:
 Find  of function .

Answer

Differentiating both sides with respect to x, we obtain

Question 13:

Find 

Answer

Differentiating both sides with respect to x, we obtain

Question 14:
 Find  of function  .

Answer

Differentiating both sides, we obtain

Question 15:
 Find   of function .

Answer

Differentiating both sides with respect to x, we obtain

Question 16:
 Find the derivative of the function given by    and
 hence find .

Answer
The given relationship is
Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Question 17:
 Differentiate   in three ways mentioned below
 (i) By using product rule.
 (ii) By expanding the product to obtain a single polynomial.
 (iii By logarithmic differentiation.
 Do they all give the same answer?

Answer

Let y =
(i)

(ii)

( iii)

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

From the above three observations, it can be concluded that all the results of   are same.

 Question 18: If u, v and w are functions of x, then show that

in two ways-first by repeated application of product rule, second by logarithmic
 differentiation.

Answer
Let
By applying product rule, we obtain