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 waysfirst by repeated application of product rule, second by logarithmic
differentiation.
Answer
Let
By applying product rule, we obtain
1. What is continuity in calculus? 
2. What does differentiability mean in calculus? 
3. How are continuity and differentiability related? 
4. How can we determine if a function is continuous? 
5. How do we test for differentiability of a function? 
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129 videos359 docs306 tests
