Master Calculus I Derivatives: 20 Question Quiz

Question 1

Find the derivative of $f(x) = 3x^5 - 2x^3 + 7x - 4$.

Accepted Answer

$f'(x) = 15x^4 - 6x^2 + 7$

Question 2

Find $\frac{d}{dx}[\sin(x) \cdot e^x]$.

Accepted Answer

$e^x(\sin x + \cos x)$

Question 3

Find the derivative of $g(x) = \frac{x^2 + 1}{x - 3}$.

Accepted Answer

$\frac{x^2 - 6x - 1}{(x-3)^2}$

Question 4

Find $\frac{d}{dx}[\cos(3x^2)]$.

Accepted Answer

$-6x\sin(3x^2)$

Question 5

Find the derivative of $h(x) = \ln(x^2 + 5)$.

Accepted Answer

$\frac{2x}{x^2 + 5}$

Question 6

If $y = (2x + 1)^4$, find $\frac{dy}{dx}$.

Accepted Answer

$8(2x + 1)^3$

Question 7

Find $\frac{d}{dx}[e^{\sin x}]$.

Accepted Answer

$e^{\sin x} \cos x$

Question 8

The equation of the tangent line to $f(x) = x^3$ at $x = 2$ is:

Accepted Answer

$y = 12x - 16$

Question 9

Find $\frac{d}{dx}[	an x]$.

Accepted Answer

$\sec^2 x$

Question 10

If $f(x) = \sqrt{x^3 + 2x}$, find $f'(x)$.

Accepted Answer

$\frac{3x^2 + 2}{2\sqrt{x^3 + 2x}}$

Question 11

Find $f'(2)$ if $f(x) = x^4 - 3x^2 + 5x$. Round to the nearest whole number.

Accepted Answer

29

Question 12

Find the slope of the tangent line to $y = \sin(x)$ at $x = \frac{\pi}{3}$. Round to 2 decimal places.

Accepted Answer

0.50

Question 13

If $g(x) = e^{2x}$, find $g'(0)$. Enter your answer as a whole number.

Accepted Answer

2

Question 14

Find $\frac{dy}{dx}$ at the point $(1, 2)$ if $x^2 + xy = 3$. Round to 2 decimal places.

Accepted Answer

-1.33

Question 15

Find the value of $x$ where the function $f(x) = x^3 - 6x^2 + 9x + 1$ has a horizontal tangent line. Enter the smaller value of $x$.

Accepted Answer

1

Question 16

Explain the product rule for differentiation and use it to find the derivative of $f(x) = x^2 \ln(x)$.

Question 17

Find $\frac{dy}{dx}$ using implicit differentiation: $x^2 + y^2 = 25$. Then find the slope of the tangent line at the point $(3, 4)$.

Question 18

Explain why the chain rule is necessary and use it to differentiate $f(x) = \sin(x^3)$. Show all steps.

Question 19

Find the equation of the tangent line to $f(x) = e^x$ at $x = 0$. Explain each step of your process.

Question 20

Differentiate $f(x) = \frac{\sin x}{x}$ using the quotient rule. Simplify your answer.