Quotient rule review

Review your knowledge of the Quotient rule for derivatives, and use it to solve problems.

What is the Quotient rule?

The Quotient rule tells us how to differentiate expressions that are the quotient of two other, more basic, expressions:

\frac{d}{d x} [\frac{f (x)}{g (x)}] = \frac{\frac{d}{d x} [f (x)] \cdot g (x) - f (x) \cdot \frac{d}{d x} [g (x)]}{[g (x)]^{2}}

Basically, you take the derivative of

f

multiplied by

g

, subtract

f

multiplied by the derivative of

g

, and divide all that by

[g (x)]^{2}

Want to learn more about the Quotient rule? Check out this video.

Consider the following differentiation of

\frac{\sin (x)}{x^{2}}

\begin{aligned} \frac{d}{d x} (\frac{\sin (x)}{x^{2}}) \\ = \frac{\frac{d}{d x} (\sin (x)) x^{2} - \sin (x) \frac{d}{d x} (x^{2})}{(x^{2})^{2}} & Quotient rule \\ = \frac{\cos (x) \cdot x^{2} - \sin (x) \cdot 2 x}{(x^{2})^{2}} & Differentiate \sin (x) and x^{2} \\ = \frac{x (x \cos (x) - 2 \sin (x))}{x^{4}} & Simplify \\ = \frac{x \cos (x) - 2 \sin (x)}{x^{3}} & Cancel common factors \end{aligned}

Problem 1

f (x) = \frac{x^{2}}{e^{x}}

f^{'} (x) =

Want to try more problems like this? Check out this exercise.

Suppose we are given this table of values:

$x$ ‍	$f (x)$ ‍	$g (x)$ ‍	$f^{'} (x)$ ‍	$g^{'} (x)$ ‍
$4$ ‍	$- 4$ ‍	$- 2$ ‍	$0$ ‍	$8$ ‍

H (x)

is defined as

\frac{f (x)}{g (x)}

, and we are asked to find

H^{'} (4)

The Quotient rule tells us that

H^{'} (x)

\frac{f^{'} (x) g (x) - f (x) g^{'} (x)}{[g (x)]^{2}}

. This means

H^{'} (4)

\frac{f^{'} (4) g (4) - f (4) g^{'} (4)}{[g (4)]^{2}}

. Now let's plug the values from the table in the expression:

\begin{aligned} H^{'} (4) & = \frac{f^{'} (4) g (4) - f (4) g^{'} (4)}{[g (4)]^{2}} \\ = \frac{(0) (- 2) - (- 4) (8)}{(- 2)^{2}} \\ = \frac{32}{4} \\ = 8 \end{aligned}

Problem 1

$x$ ‍	$g (x)$ ‍	$h (x)$ ‍	$g^{'} (x)$ ‍	$h^{'} (x)$ ‍
$- 2$ ‍	$4$ ‍	$1$ ‍	$- 1$ ‍	$2$ ‍

F (x) = \frac{g (x)}{h (x)}

F^{'} (- 2) =

Want to try more problems like this? Check out this exercise.

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