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**Continuity & Differentiability**

** Question 1: Differentiate the functions with respect to x.**

**Answer**

**Question 2: Differentiate the functions with respect to x. cos(sin x)**

**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(a*x* + 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

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