Product rule review

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

What is the Product rule?

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

\frac{d}{d x} [f (x) \cdot g (x)] = \frac{d}{d x} [f (x)] \cdot g (x) + f (x) \cdot \frac{d}{d x} [g (x)]

Basically, you take the derivative of

f

multiplied by

g

, and add

f

multiplied by the derivative of

g

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

Consider the following differentiation of

h (x) = \ln (x) \cos (x)

\begin{aligned} h^{'} (x) \\ = \frac{d}{d x} (\ln (x) \cos (x)) \\ = \frac{d}{d x} (\ln (x)) \cos (x) + \ln (x) \frac{d}{d x} (\cos (x)) & Product rule \\ = \frac{1}{x} \cdot \cos (x) + \ln (x) \cdot (- \sin (x)) & Differentiate \ln (x) and \cos (x) \\ = \frac{\cos (x)}{x} - \ln (x) \sin (x) & Simplify \end{aligned}

Problem 1

f (x) = 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:

$x$ ‍	$f (x)$ ‍	$g (x)$ ‍	$f^{'} (x)$ ‍	$g^{'} (x)$ ‍
$4$ ‍	$- 4$ ‍	$13$ ‍	$0$ ‍	$8$ ‍

H (x)

is defined as

f (x) \cdot g (x)

, and we are asked to find

H^{'} (4)

The Product rule tells us that

H^{'} (x)

f^{'} (x) g (x) + f (x) g^{'} (x)

. This means

H^{'} (4)

f^{'} (4) g (4) + f (4) g^{'} (4)

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

\begin{aligned} H^{'} (4) & = f^{'} (4) g (4) + f (4) g^{'} (4) \\ = (0) (13) + (- 4) (8) \\ = - 32 \end{aligned}

Problem 1

$x$ ‍	$g (x)$ ‍	$h (x)$ ‍	$g^{'} (x)$ ‍	$h^{'} (x)$ ‍
$- 2$ ‍	$2$ ‍	$- 1$ ‍	$3$ ‍	$4$ ‍

F (x) = g (x) \cdot h (x)

F^{'} (- 2) =

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

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