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

## EXERCISE 5.2

#### Question 1:

Differentiate the functions with respect to x. 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.  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.  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.  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. 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. The given function is . #### Question 7:

Differentiate the functions with respect to x.  #### Question 8:

Differentiate the functions with respect to x.  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.

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.

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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