## 1. If a function is differentiable at a point, it must be:

### Answer:

### Explanation:

Differentiability at a point implies continuity at that point.

## 2. Which of the following functions is not continuous at x = 0?

### Answer:

### Explanation:

tan(x) is undefined at x = π/2 and its multiples, including x = 0.

## 3. The derivative of a constant is:

### Answer:

### Explanation:

Constants don't change, so their rate of change (derivative) is zero.

## 4. If f(x) = |x|, then f is:

### Answer:

### Explanation:

The modulus function has a sharp turn at x = 0, making it non-differentiable there.

## 5. The derivative of f(x) = x^3 with respect to x is:

### Answer:

### Explanation:

Using the power rule for differentiation.

## 6. A function which is continuous in its domain:

### Answer:

### Explanation:

Continuity doesn't guarantee differentiability.

## 7. The derivative of tan(x) is:

### Answer:

### Explanation:

Basic differentiation rule for tan(x).

## 8. If f(x) = ln(x), its derivative is:

### Answer:

### Explanation:

Basic differentiation rule for natural logarithm.

## 9. A function is said to be differentiable at x = a if:

### Answer:

### Explanation:

Differentiability requires the function's slope from the left and right to be the same at the point.

## 10. If g(x) = e^x, its derivative is:

### Answer:

### Explanation:

The exponential function is its own derivative.

## 11. The point where the function changes its nature from increasing to decreasing or vice-versa is:

### Answer:

### Explanation:

At critical points, the derivative is either zero or does not exist.

## 12. If a function is differentiable in (a, b), then it is:

### Answer:

### Explanation:

Differentiability implies continuity, but not vice-versa.

## 13. The derivative of sin^2(x) with respect to x is:

### Answer:

### Explanation:

Using the chain rule and the derivative of sin(x).

## 14. If h(x) = 1/x, h is not differentiable at:

### Answer:

### Explanation:

1/x is undefined at x = 0.

## 15. The second derivative measures:

### Answer:

### Explanation:

The second derivative gives insight into the concavity or convexity of a function.

## 16. The value of d/dx [x^x] at x = 1 is:

### Answer:

### Explanation:

Differentiating x^x and evaluating at x = 1 gives the result.

## 17. The derivative of f(x) = log(x) to the base a (where a > 0, a ≠ 1) is:

### Answer:

### Explanation:

Differentiation rule for logarithms with bases other than e.

## 18. A function which is differentiable on its domain:

### Answer:

### Explanation:

Differentiability ensures a smooth curve.

## 19. The derivative of the function f(x) = √x is:

### Answer:

### Explanation:

Using the power rule for differentiation.

## 20. If y = u/v and both u and v are differentiable functions of x, then dy/dx is:

### Answer:

### Explanation:

Using the quotient rule for differentiation.