JEE  >  NCERT Solutions Exercise 5.5: Continuity & Differentiability

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

Continuity & Differentiability

Question 1: Differentiate the function with respect to x.

Question 2: Differentiate the function with respect to x.

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.

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. xx - 2sinx

Let y =

u = xx
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.

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.

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.

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.

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.

Let y =

Differentiating both sides with respect to x, we obtain

Question 10: Differentiate the function with respect to x.

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.

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

Question 12:
Find
of function .

Differentiating both sides with respect to x, we obtain

Question 13:

Find

Differentiating both sides with respect to x, we obtain

Question 14:
Find  of function  .

Differentiating both sides, we obtain

Question 15:
Find
of function .

Differentiating both sides with respect to x, we obtain

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

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?

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.

Let
By applying product rule, we obtain

The document NCERT Solutions Exercise 5.5: Continuity & Differentiability | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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## FAQs on NCERT Solutions Exercise 5.5: Continuity & Differentiability - Mathematics (Maths) for JEE Main & Advanced

 1. What is continuity in calculus?
Ans. Continuity refers to the property of a function where there are no abrupt jumps, breaks, or holes in the graph. It means that the function is smooth and can be drawn without lifting the pen from the paper. In simpler terms, a function is continuous if its graph can be traced without any interruptions.
 2. What does differentiability mean in calculus?
Ans. Differentiability is a property of a function where it has a derivative at every point in its domain. It implies that the function is smooth and its graph doesn't have sharp turns or corners. A differentiable function can be approximated well by a straight line at any point on its graph.
 3. How are continuity and differentiability related?
Ans. Continuity is a necessary condition for differentiability. If a function is differentiable at a point, then it must also be continuous at that point. However, continuity alone doesn't guarantee differentiability. A function can be continuous but not differentiable if there is a sharp corner or vertical tangent at a particular point.
 4. How can we determine if a function is continuous?
Ans. To determine if a function is continuous, we need to check three conditions: 1. The function should be defined at the given point. 2. The limit of the function as it approaches the given point should exist. 3. The value of the function at the given point should be equal to the limit. If all three conditions are satisfied, then the function is continuous at that point.
 5. How do we test for differentiability of a function?
Ans. To test for differentiability of a function, we need to check if the derivative exists at each point in its domain. The derivative can be calculated using differentiation rules or formulas. If the derivative exists for all points in the domain, then the function is differentiable. However, if there are any points where the derivative doesn't exist, then the function is not differentiable at those points.

## Mathematics (Maths) for JEE Main & Advanced

129 videos|359 docs|306 tests

## Mathematics (Maths) for JEE Main & Advanced

129 videos|359 docs|306 tests

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