NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

The document NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.
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Continuity & Differentiability 

Question 1: Differentiate the function with respect to x.

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev


Answer
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

Question 2: Differentiate the function with respect to x.

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev


Answer

Let y = NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Taking logarithm on both the sides, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

Differentiating both sides with respect to x, we obtain 

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

Question 3: Differentiate the function with respect to x. NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

Answer

Let y = NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Taking logarithm on both the sides, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

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


Answer

Let y = NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
u = xx
Taking logarithm on both the sides, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

 

Question 5: Differentiate the function with respect to x.

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

Answer

Let y = NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Taking logarithm on both the sides, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

 


Question 6: Differentiate the function with respect to x.

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev


Answer

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Therefore, from (1), (2), and (3), we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev



Question 7: Differentiate the function with respect to x.

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

Answer

Let y = NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev


Question 8: Differentiate the function with respect to x.

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

Answer

Let y = NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

Differentiating both sides with respect to x, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Therefore, from (1), (2), and (3), we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev


Question 9: Differentiate the function with respect to x.

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev


Answer
Let y = NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

Question 10: Differentiate the function with respect to x.

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev


Answer

Let y = NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev


Question 11: Differentiate the function with respect to x.
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

Answer

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
 

Question 12:
 Find 
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev of function NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev.


Answer
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev


Question 13:

Find NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev


Answer
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev



Question 14:
 Find NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev of function NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev .


Answer
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev



Question 15:
 Find 
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev  of function NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev.


Answer
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev


Question 16:
 Find the derivative of the function given by  
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev  and
 hence find 
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev.


Answer
The given relationship is NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Taking logarithm on both the sides, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Question 17:
 Differentiate 
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev  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 = NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
(i)

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev


(ii)

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
 


( iii) NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

Taking logarithm on both the sides, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
Differentiating both sides with respect to x, we obtain

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

From the above three observations, it can be concluded that all the results of NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev  are same.

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

NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
in two ways-first by repeated application of product rule, second by logarithmic
 differentiation.


Answer
Let NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev
By applying product rule, we obtain
NCERT Solutions - Continuity & Differentiability, Exercise 5.5 JEE Notes | EduRev

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