Continuity And Differentiability

Which of the following statements is always true?

1. Every continuous function is differentiable
2. Every differentiable function is continuous
3. Every discontinuous function is differentiable
4. Every constant function is discontinuous

If f(x) = sin inverse of x, then f'(x) is

1. 1/sqrt(1 - x squared)
2. 1/(1 + x squared)
3. 1/sqrt(x squared - 1)
4. sec squared x

If y squared + x squared = 25, then dy/dx is

1. -x/y
2. y/x
3. x/y
4. -y/x

The derivative of f(x) = log base a of x is

1. 1/x
2. 1/(x ln a)
3. ln x
4. a^x

If f(x) = tan x, then f'(x) is

1. sec x
2. sec squared x
3. cosec squared x
4. cot x

If a function is differentiable at a point, then it is necessarily

1. Discontinuous
2. Continuous
3. Constant
4. Increasing

The derivative of f(x) = ln x is

1. x
2. 1/x
3. ln x
4. e^x

If y = (3x + 2)^5, then dy/dx is

1. 5(3x + 2)^4
2. 15(3x + 2)^4
3. 15(3x + 2)^5
4. 3(3x + 2)^4

The derivative of f(x) = (x squared + 1)/(x + 1) is obtained by using

1. Product Rule
2. Quotient Rule
3. Chain Rule
4. Power Rule

If f(x) = x squared sin x, then f'(x) is

1. 2x sin x
2. x squared cos x
3. 2x sin x + x squared cos x
4. 2x cos x + x squared sin x

If f(x) = x cubed, then f'(x) is

1. x squared
2. 2x
3. 3x squared
4. 3x

The derivative of f(x) = e^x is

1. xe^(x-1)
2. e^x
3. 1/e^x
4. xe^x

A function f(x) is continuous at x = a if

1. Only f(a) exists
2. Only the limit exists
3. Left-hand limit = Right-hand limit = f(a)
4. Left-hand limit = Right-hand limit only

The derivative of f(x) = sin x is

1. -sin x
2. cos x
3. -cos x
4. tan x

If f(x) = x squared + 3x + 2, then the value of f'(2) is

1. 5
2. 7
3. 9
4. 11