Continuity & Differentiability
Question 1:
Answer
Question 2:
Answer
Question 3:
Answer
Question 4:
Answer
The given relationship is 
Differentiating this relationship with respect to x, we obtain
Question 5:
Find 
Answer
The given relationship is x2 +xy +y2 = 100
Differentiating this relationship with respect to x, we obtain
[Derivative of constant function is 0]
Question 6:
Answer
Question 7:
Answer
Using chain rule, we obtain
Question 8:
Find 
Answer
The given relationship is
Differentiating this relationship with respect to x, we obtain
Question 9:
Find 
Answer
Therefore, by quotient rule, we obtain
Question 10:
Answer
Question 11:
Answer
The given relationship is,
On comparing L.H.S. and R.H.S. of the above relationship, we obtain tany/2 = x
Differentiating this relationship with respect to x, we obtain
Question 12:
Answer
The given relationship is 
From (1), (2), and (3), we obtain
Differentiating this relationship with respect to x, we obtain
Question 13:
Answer
Question 14:
Find 
Answer
Differentiating this relationship with respect to x, we obtain
Question 15:
Find dy/dx 
Answer
Differentiating this relationship with respect to x, we obtain
FAQs on NCERT Solutions Class 12 Maths Chapter 5 - Continuity & Differentiability
| 1. What is continuity and differentiability in calculus? |  |
Ans. Continuity and differentiability are important concepts in calculus. Continuity refers to the smoothness of a function, where there are no abrupt changes or breaks in its graph. A function is said to be continuous at a point if the limit of the function exists at that point and is equal to the value of the function at that point.
Differentiability, on the other hand, deals with the concept of the derivative of a function. A function is said to be differentiable at a point if the derivative exists at that point. The derivative represents the rate of change of the function at a particular point and provides information about the slope of the tangent line to the graph of the function at that point.
| 2. What are the conditions for a function to be continuous at a point? | |
Ans. For a function to be continuous at a point, three conditions must be satisfied:
1. The function must be defined at that point.
2. The limit of the function as x approaches that point must exist.
3. The value of the function at that point must be equal to the limit.
If any of these conditions are not met, the function is said to be discontinuous at that point.
| 3. How do you determine if a function is differentiable at a point? | |
Ans. To determine if a function is differentiable at a point, we need to check if the derivative exists at that point. The derivative of a function represents the rate of change of the function at a particular point.
To find the derivative at a point, we can use the definition of the derivative or apply differentiation rules, such as the power rule or chain rule, depending on the complexity of the function. If the derivative exists at the given point, then the function is differentiable at that point.
| 4. Can a function be differentiable but not continuous? | |
Ans. No, a function cannot be differentiable at a point if it is not continuous at that point. Differentiability requires the function to be continuous at the point of interest.
If a function has a discontinuity, such as a jump or an essential discontinuity, it cannot have a derivative at that point. The existence of a derivative depends on the smoothness and continuity of the function.
| 5. How are continuity and differentiability related? | |
Ans. Continuity and differentiability are closely related concepts in calculus. A function must be continuous at a point for it to be differentiable at that point.
If a function is continuous at a point, it means that there are no abrupt changes or breaks in its graph at that point. This smoothness allows us to define a derivative, which represents the rate of change of the function at that point.
In simpler terms, differentiability implies continuity, but continuity does not necessarily imply differentiability. A function can be continuous at a point without being differentiable, but it cannot be differentiable at a point without being continuous.
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