Continuity & Differentiability
Question 1: Differentiate the functions with respect to x.
Answer
Question 2: Differentiate the functions with respect to x. cos(sinx)
Answer
Thus, f is a composite function of two functions.
Put t = u (x) = sin x
By chain rule,
Alternate method
Question 3: Differentiate the functions with respect to x.
sin(ax + b)
Answer
Alternate method
Question 4: Differentiate the functions with respect to x.
Answer
Hence, by chain rule, we obtain
Question 5: Differentiate the functions with respect to x.
Answer
The given function is
Put y = p (x) = cx + d
Question 6: Differentiate the functions with respect to x. 
Answer
Question 7: Differentiate the functions with respect to x.
Answer
Question 8: Differentiate the functions with respect to x. 
Answer
Clearly, f is a composite function of two functions, u and v, such that
Alternate method
Question 9:
Prove that the function f given by 
is not differentiable at x = 1.
Answer
The given function is 
It is known that a function f is differentiable at a point x = c in its domain if both
are finite and equal.
To check the differentiability of the given function at x = 1,
consider the left hand limit of f at x = 1
Since the left and right hand limits of f at x = 1 are not equal, f is not differentiable at x = 1
Question 10:
Prove that the greatest integer function 
defined by is not differentiable at x = 1 and x = 2.
Answer
The given function f is 
It is known that a function f is differentiable at a point x = c in its domain if both
are finite and equal.
To check the differentiability of the given function at x = 1, consider the left hand limit of f at x = 1
Since the left and right hand limits of f at x = 1 are not equal, f is not differentiable at x = 1
To check the differentiability of the given function at x = 2, consider the left hand limit of f at x = 2
Since the left and right hand limits of f at x = 2 are not equal, f is not differentiable at x = 2
FAQs on NCERT Solutions Class 12 Maths Chapter 5 - Continuity & Differentiability
| 1. What is continuity in mathematics? |  |
Ans. Continuity in mathematics refers to the property of a function where it does not have any abrupt changes or discontinuities. Essentially, it means that the function 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 breaks or holes.
| 2. What is differentiability in calculus? | |
Ans. Differentiability in calculus refers to the property of a function where it has a derivative at every point in its domain. A function is said to be differentiable at a point if it has a well-defined tangent line or slope at that point. In other words, the function should not have any sharp corners or vertical tangents.
| 3. How do we check the continuity of a function? | |
Ans. To check the continuity of a function, we need to ensure three conditions:
1. The function must be defined at that point.
2. The limit of the function as it approaches that point should exist.
3. The limit of the function as it approaches that point should be equal to the value of the function at that point.
If all three conditions are satisfied, the function is continuous at that point. If any of the conditions fail, the function is discontinuous at that point.
| 4. What is the difference between continuity and differentiability? | |
Ans. Continuity and differentiability are related concepts in calculus but have distinct meanings. Continuity refers to the absence of abrupt changes or discontinuities in a function, whereas differentiability refers to the existence of a derivative at every point in the function's domain.
In simpler terms, a function can be continuous without being differentiable, but a function cannot be differentiable without being continuous. Differentiability is a stronger condition than continuity.
| 5. Can a function be continuous but not differentiable? | |
Ans. Yes, a function can be continuous but not differentiable. This occurs when a function has a sharp corner, vertical tangent, or a point where the derivative does not exist. These points are known as points of non-differentiability. So, while continuity ensures the absence of abrupt changes, differentiability requires the function to have a well-defined tangent or derivative at every point.
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