The limit $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$, provided the limit exists.

Question 1

Which of the following best describes the derivative of a function $$f(x) at a $$point $$x$$?

Accepted Answer

The limit $$f'(x) = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$$, provided the limit exists.

Question 2

For which of the following functions is the derivative NOT defined at $$x = 0$$?

Accepted Answer

$$f(x) = |x|$$

Question 3

What is the derivative of the constant function $$f(x) = 7$$?

Accepted Answer

$$0$$

Question 4

Which rule allows you to find the derivative of $$f(x) = 5x^3$?

Accepted Answer

Constant Multiple Rule and Power Rule

Question 5

If $$u(x)$$ and $$v(x)$$ are differentiable, what is the derivative of their product $$u(x)v(x)$$?

Accepted Answer

$$u(x) \frac{dv}{dx} + v(x) \frac{du}{dx}$$

Question 6

Which of the following is true about the differentiability of $$f(x) = \sqrt{x} at$$ $$x = 0$$?

Accepted Answer

It is not differentiable at $$x = 0$$ due to a vertical tangent.

Question 7

Given $$f(x) = x^n$$ for any real number $$n$$, what is $$\frac{d}{dx} x^n$$?

Accepted Answer

$$n x$$^{n-1}$$

Question 8

If $$f(x) = 3x^2$ + 2x + 5$$, what is $$f'(x)$$?

Accepted Answer

$$6x + 2$$

Question 9

Which of the following is the correct application of the quotient rule for $$f(x) = \frac{u(x)}{v(x)}$$?

Accepted Answer

$$\frac{v(x) \frac{du}{dx} - u(x) \frac{dv}{dx}}{[v(x)]^2}$$

Question 10

A function $$f(x)$$ has a 'cusp' at $$x = a if$$:

Accepted Answer

The slope approaches $$+\infty$$ from one side and $$-\infty$$ from the other.

Question 11

If $$f(x) = x^3$ - 2x$$, what is $$f'(x)$$?

Accepted Answer

$$3x^2 - 2$

Question 12

Which of the following is NOT a reason for a function to be non-differentiable at a point?

Accepted Answer

The function is continuous and smooth at the point.

Question 13

If $$f(x) = 2x^2$ and g(x) = x + 1$, what is the derivative of f(x) + g(x$$)$$?

Accepted Answer

$$4x + 1$$

Question 14

A real-world example: The position of a car is given by $$s(t) = 4t^2$ + 2t + 1. $$What is the instantaneous velocity at time $$t$$?

Accepted Answer

$$8t + 2$$

Question 15

If $$f(x) = x^4$ and g(x$$) = $$x^2$$, what is the derivative of $$f(x)g(x)$$?

Accepted Answer

$$x^2$$ \cdot $$4x^3$$ + $$x^4$$ \cdot 2x$$

Question 16

Which of the following statements about the derivative of a constant function is correct?

Accepted Answer

The derivative is always $$0.$$

Question 17

If $$f(x) = 2x^3$ - 5x + 4$$, what is $$f'(x)$$?

Accepted Answer

$$6x^2 - 5$

Question 18

Which rule would you use to differentiate $$h(x) = \frac{3x^2$ + 2x}{x}$$?

Accepted Answer

Quotient Rule

Question 19

If $$f(x) = x^2$ and g(x$$) = $$3x^2$$, how does the slope of $$g(x)$$ compare to $$f(x) at$$ $$x = 1$$?

Accepted Answer

It is three times the slope of $$f(x) at$$ $$x = 1.$$

Section 3.3: Differentiation Rules – Comprehensive Study Notes

Study Guide - Practice Questions