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NCERT solution class 12 chapter 5 Continuity and Differentiability exercise 5.2 mathematics part 1

EXERCISE 5.2


Page No 166:

Question 1:

Differentiate the functions with respect to x.

Answer:

Let f(x)=sinx2+5, ux=x2+5, and v(t)=sint

 

Then, vou=vux=vx2+5=tanx2+5=f(x)

Thus, f is a composite of two functions.

Alternate method

Question 2:

Differentiate the functions with respect to x.

Answer:

Thus, 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.

Answer:

Thus, is a composite function of two functions, u and v.

Put t = u (x) = ax + b

Hence, by chain rule, we obtain

Alternate method

Question 4:

Differentiate the functions with respect to x.

Answer:

Thus, is a composite function of three functions, u, v, and w.

Hence, by chain rule, we obtain

Alternate method

Question 5:

Differentiate the functions with respect to x.

Answer:

The given function is, where g (x) = sin (ax + b) and

h (x) = cos (cx d)

∴ is a composite function of two functions, u and v.

Therefore, by chain rule, we obtain

h is a composite function of two functions, p and q.

Put y = p (x) = cx d

Therefore, by chain rule, we obtain

Question 6:

Differentiate the functions with respect to x.

Answer:

The given function is.

Question 7:

Differentiate the functions with respect to x.

Answer:

Question 8:

Differentiate the functions with respect to x.

Answer:

Clearly, is a composite function of two functions, and v, such that

By using chain rule, we obtain

Alternate method

Question 9:

Prove that the function given by

 is notdifferentiable 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 byis 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


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