Free Practice Quiz Question List

3 The Derivative and Differentiation Rules Online Quiz Questions

Use this free practice quiz with 20 questions to review 3 The Derivative and Differentiation Rules, test your knowledge, and prepare for your next test or exam.

20 questions

Choose one

1 point

Which interpretation of $f'(a)$ is correct?

A
The slope of a secant line over a nonzero interval
B
The slope of the tangent line at a point
C
The height of the function at that point
D
The area between the graph and the axis

Choose one

1 point

Which statement about differentiability and continuity is supported by the material?

A
If a function is differentiable at a point, then it is continuous there.
B
If a function is continuous at a point, then it must be differentiable there.
C
A function is differentiable at a point exactly when its value is zero there.
D
Differentiability and continuity are unrelated properties.

Choose one

1 point

If $s(t)$ is position measured in meters and $t$ is time measured in seconds, what are the units of $s'(t)$ ?

A
Seconds per meter
B
Meters
C
Meters per second
D
Square meters per second

Choose all

1 point

Select all formulas that correctly state a differentiation rule.

A
$\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$
B
$\frac{d}{dx}[f(x)g(x)]=f'(x)g'(x)$
C
$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}$
D
$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f'(x)}{g'(x)}$

Choose all

1 point

Select all statements that correctly apply the constant or power rule.

A
$\frac{d}{dx}[7]=7$
B
$\frac{d}{dx}[7]=0$
C
$\frac{d}{dx}[x^5]=x^4$
D
$\frac{d}{dx}[x^5]=5x^4$

True or false

1 point

True or false: The derivative at $a$ is defined by $\displaystyle f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$ , provided the limit exists.

A
True
B
False

True or false

1 point

True or false: Because $f(x)=|x|$ is continuous at $x=0$ , it is differentiable there.

A
True
B
False

Written response

1 point

For $f(x)=3x^4-2x+7$ , enter the value of $f'(2)$ .

Fill in the blank

1 point

To differentiate a composition such as $\sin(x^2)$ , use the . First differentiate the outer function while keeping the unchanged, then multiply by its derivative.

Fill in the blank

1 point

For the relation $xy+y^2=6$ , enter the simplified result of implicit differentiation in the blank: .

Open ended

1 point

Using the chain rule, find $\frac{d}{dx}(3x^2-5)^4$ . Show the outer and inner derivatives in your reasoning.

Choose one

1 point

Which expression is the derivative of $x^2\sin x$ ?

A
$2x\sin x+x^2\cos x$
B
$2x\cos x$
C
$x^2\cos x$
D
$2x\sin x$

Choose one

1 point

What does $f'(a)$ represent geometrically when the derivative exists?

A
The average value of the function near the point
B
The slope of the tangent line at the point
C
The vertical distance from the graph to the x-axis
D
The x-coordinate where the function is undefined

Choose one

1 point

Using the quotient rule, which expression gives the derivative of $\frac{x^2+1}{x-1}$ ?

A
$\frac{(x-1)(1)-(x^2+1)(2x)}{(x-1)^2}$
B
$\frac{2x}{x-1}$
C
$\frac{(x-1)(2x)-(x^2+1)}{(x-1)^2}$
D
$\frac{(x^2+1)(2x)-(x-1)}{(x-1)^2}$

Choose one

1 point

What is $\frac{d}{dx}\left[(3x^2-5)^4\right]$ ?

A
$4(3x^2-5)^3$
B
$12x(3x^2-5)^4$
C
$24x(3x^2-5)^4$
D
$24x(3x^2-5)^3$

Choose one

1 point

The circle $x^2+y^2=25$ contains the point $(3,4)$ . What is the slope $\frac{dy}{dx}$ at that point?

A
$\frac{3}{4}$
B
$-\frac{3}{4}$
C
$-\frac{4}{3}$
D
$\frac{4}{3}$

True or false

1 point

True or false: If a function is differentiable at a point, then it must be continuous at that point.

A
True
B
False

Written response

1 point

Using the chain rule, find $\frac{d}{dx}[\sin(5x^3)]$ . Enter the result as a simplified product in LaTeX, with the factors $15x^2$ and $\cos(5x^3)$ .

Written response

1 point

For the relation $xy+y^2=6$ , find $\frac{dy}{dx}$ in terms of $x$ and $y$ . Enter the answer in the canonical form of one simplified fraction: a negative fraction with $y$ in the numerator and $x+2y$ in the denominator, using LaTeX mathematical delimiters.

Written response

1 point

Let $f(x)=\arctan x$ . What is $f'(1)$ ?