Free Printable Calc: Quotient and Chain Rule Quiz

1.

What is NOT an acceptable way to write the quotient rule for the derivative of y = $\frac{f}{g}$

a)

$y'=\frac{(gf'-fg')}{g^2}$

b)

$y'=\frac{(f'g-g'f)}{g^2}$

c)

$y'=\frac{lo\cdot dhi-hi\cdot dlo}{lo^2}$

d)

$y'=\frac{(fg'-gf')}{g^2}$

2.

True or False:

\frac{d}{dx}\left(\left(f\left(g\left(x\right)\right)\right)\right)=f'\left(g'\left(x\right)\right)

a)

True

b)

False

c)

Can't tell, it depends on the rule

d)

It's all the parentheses for me

3.

Which rule can you NOT use the chain rule on?

a)

$\left(8x^3+π\right)^{10}$

b)

$\ln\left(5x^3-3\right)$

c)

$\tan\left(400x^2\right)$

d)

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

4.

Which rule can you NOT use the quotient rule on? $\$

a)

$\cot x$

b)

$\frac{e^x}{\sec x}$

c)

$\ln\left(4x\right)$

d)

Chai $\frac{x+6}{6x^2}$ n Rule

5.

Which of the following doesn't require the quotient rule? Meaning it can be rewritten and differentiated with a different rule.

a)

$\frac{\sin x}{x^2}$

b)

$\frac{x^4-5x+6}{4x^3+2x}$

c)

$\frac{\ln x}{e^x}$

d)

$\frac{5}{x^4}$

6.

$y=\csc^8x$ Which of the following is the proper way to setup up the chain rule? $\$

a)

$y=u^8,\ u\ =\ \csc x$

b)

$y=\csc u,\ u\ =\ x^8$

c)

Not a composite function

7.

Set up the derivative of $y\left(x\right)=\frac{3x-7}{\left(x^2+2x+1\right)}$

a)

$y'\left(x\right)=\frac{\left(3x-7\right)\left(2x+2\right)-\left(3\right)\left(x^2+2x+1\right)}{\left(3x-7\right)^2}$

b)

$y'\left(x\right)=\frac{\left(3x-7\right)\left(2x+2\right)-3\left(x^2+2x+1\right)}{\left(x^2+2x+1\right)^2}$

c)

$y'\left(x\right)=\frac{\left(3\right)\left(x^2+2x+1\right)-\left(3x-7\right)\left(2x+2\right)}{\left(x^2+2x+1\right)}$

d)

$y'\left(x\right)=\frac{\left(3\right)\left(x^2+2x+1\right)-\left(3x-7\right)\left(2x+2\right)}{\left(x^2+2x+1\right)^2}$

8.

Find and simplify y' if $y=\frac{4x^2-5}{x+3}$

a)

$y'=\frac{8x^2+8x+5}{\left(x+3\right)^2}$

b)

$y'=\frac{4x^2+24x+5}{\left(x+3\right)^2}$

c)

$y'=\frac{4x^2+24x-5}{\left(x+3\right)^2}$

d)

$y'=\frac{8x^2+24x+5}{\left(4x^2-5\right)^2}$

9.

Find AND simplify the derivative of $\frac{2x^4}{\sec x}$

a)

$y'\left(x\right)=\frac{x\tan x+4}{\sec x}$

b)

$y'\left(x\right)=\frac{2x^3\left(4-x\tan x\right)}{\sec x}$

c)

$y'\left(x\right)=\frac{2x^3\sec x\left(4-x\tan x\right)}{\left(2x^4\right)^2}$

d)

$y'\left(x\right)=\frac{8x^3\sec x-2x^4\sec x\tan x}{\sec^2x}$

10.

Find AND fully simplify the derivative of $y'\left(x\right)=\frac{-\sin x}{e^x}\$

a)

$y'\left(x\right)=\frac{-\cos x\cdot e^x+\sin x\cdot e^x}{\left(e^x\right)^2}$

b)

$y'\left(x\right)=\frac{\cos x-\sin x}{e^x}$

c)

$y'\left(x\right)=\frac{-\cos x+\sin x}{e^x}$

d)

$y'\left(x\right)=\frac{-\sin x\cos x-e^{2x}}{\left(-\sin x^2\right)}$

11.

Find the derivative of f(x) = (x⁶ + 4)⁵

a)

f '(x) = 5x⁵(x⁴ + 4)⁴

b)

f '(x) = 6x⁵(x⁶ + 4)⁴

c)

f '(x) = 30x⁵(x⁶ + 4)⁴

d)

f '(x) = 30x⁶(x⁶ + 4)⁴

12.

Find the derivative of $\tan\ \left(10x^5\right)$

a)

$\sec^2\left(10x^5\right)$

b)

$\sec^2\left(50x^4\right)$

c)

$50x^4\tan\left(10x^5\right)$

d)

$50x^4\sec^2\left(10x^5\right)$

13.

What is the derivative of $\ln\left(x^4\right)$

a)

$\frac{4}{x}$

b)

$\frac{1}{x^4}$

c)

$\frac{x^4}{1}$

d)

$\frac{1}{4}\ln\left(x^4\right)$

14.

What is the derivative of $e^{\csc x}$ ?

a)

$-\csc xe^{\csc x}$

b)

$e^{\csc x}$

c)

$\csc xe^{\csc x-1}$

d)

$-\csc x\cot xe^{\csc x}$

15.

Evaluate f'(2) given $f\left(x\right)=\left(5x^2-8x\right)^3$ Enter your answer as a whole number or simplified fraction only (no letters).

(a)

16.

Find f'(1) $f(x)=\frac{x^2-2x-1}{3x}$

(a)

17.

Given H(x) = f(x)/g(x), use the table to evaluate H'(5). Enter your answer as a whole number or simplified fraction only (no letters).

(a)

18.

Given J(x) = f(g(x)), use the table to evaluate J'(-8). Enter your answer as a whole number or simplified fraction only (no letters).

(a)

19.

Find the error in $\frac{d}{dx}\left(\frac{x^3}{\sin x}\right)=\frac{3x^2}{\cos x}$ , if there is one.

a)

They didn't use the quotient rule

b)

The numerator was differentiated incorrectly

c)

The denominator was differentated incorrectly

d)

It's correct

20.

What is the error?

a)

The inside derivative is incorrect

b)

The outside derivative is incorrect

c)

The inside/outside functions are mislabeled

d)

The wrong rule was used

21.

Find the error in $\frac{d}{dx}\left(\frac{e^x}{x^2}\right)=\frac{x^2e^x+2xe^x}{x^4}$ , if there is one.

a)

The denominator $x^4$ should be $x^2$

b)

This problem shouldn't be done with the quotient rule

c)

It should be -, not + in the numerator

d)

It's correct!

22.

What is the error?

a)

The inner derivative is missing

b)

The outer derivative is missing

c)

The inside/outside functions are mislabeled

d)

Nothing, it's correct

Calc: Quotient and Chain Rule Quiz

Total questions: 22

Worksheet time: 45mins

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