Free Practice Quiz Question List

Differentiation Online Quiz Questions

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

20 questions

Choose one

1 point

Which expression defines the derivative of a function $f$ at $x$ using an input change $h$ ?

A
The average value of f over an interval
B
$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$
C
The y-intercept of the tangent line
D
$\frac{f(x)}{x}$

Choose all

1 point

Which two rules are directly needed to differentiate the expression $(x^2+1)^5\sin x$ without first expanding it? Select all correct choices.

A
Product rule
B
Quotient rule
C
Chain rule
D
Power rule only

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

What differentiation rule is required when differentiating a composition such as $\sin(3x^2)$ ?

Choose one

1 point

What is the derivative of $(3x^2-1)^5$ ?

A
$5(3x^2-1)^4$
B
$30(3x^2-1)^4$
C
$30x(3x^2-1)^4$
D
$15x(3x^2-1)^5$

Fill in the blank

1 point

For the function $f(x)=x^2$ , the derivative is , so the slope at $x=a$ is .

Fill in the blank

1 point

For the relation $x^2+y^2=25$ , implicit differentiation gives $\frac{dy}{dx}=$ .

Open ended

1 point

Explain why $f(x)=|x|$ is continuous at $x=0$ but not differentiable there. Your explanation should use the one-sided slopes.

Choose all

1 point

Which two statements correctly give standard derivatives? Select all correct choices.

A
$\frac{d}{dx}[\cos x]=\sin x$
B
$\frac{d}{dx}[\sin x]=\cos x$
C
$\frac{d}{dx}[\ln x]=x$
D
$\frac{d}{dx}[a^x]=a^x\ln a$ , for $a>0$ and $a\ne1$

True or false

1 point

True or false: At an endpoint of a function's domain, a one-sided derivative may be used.

A
True
B
False

Choose one

1 point

What is the second derivative of $f(x)=x^4-3x^2+2x$ ?

A
$12x^2-6$
B
$4x^3-6x+2$
C
$12x^3-6x$
D
$4x^2-6$

Choose one

1 point

Which expression gives the derivative $f'(a)$ of a function at $x=a$ ?

A
$\displaystyle \lim_{h\to 0}\frac{f(a+h)-f(a)}{a}$
B
$\displaystyle \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$
C
$\displaystyle \lim_{h\to a}\frac{f(a+h)-f(a)}{h}$
D
$\displaystyle \lim_{h\to 0}\frac{f(a+h)+f(a)}{h}$

True or false

1 point

True or false: If a function is continuous at $x=a$ , then it must be differentiable at $x=a$ .

A
True
B
False

Choose one

1 point

What is the derivative of $f(x)=x^2\sin x$ ?

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

Choose one

1 point

Let $f(x)=\frac{x^2+1}{x-3}$ . What is $f'(4)$ ?

A
$-9$
B
$-7$
C
$7$
D
$9$

Choose one

1 point

If $f(x)=(3x^2-1)^5$ , what is $f'(1)$ ?

A
$30$
B
$120$
C
$240$
D
$480$

Choose one

1 point

The circle $x^2+y^2=25$ contains the point $(3,4)$ . What is the slope of the tangent line at that point?

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

Written response

1 point

Let $y=x^x$ for $x>0$ . Using logarithmic differentiation, find the numerical value of $y'$ at $x=2$ . Enter your answer rounded to three decimal places; an absolute tolerance of 0.001 is allowed.

Written response

1 point

Define $f(t)=t^3+t$ , and let $f^{-1}$ be its inverse. Find the numerical value of $(f^{-1})'(2)$ .

Written response

1 point

For $f(x)=\ln|x^2-4|$ , find the numerical value of $f'(3)$ .