The Product Rule (College Board AP® Calculus BC): Revision Note

Don't confuse the product of two functions with a composite function

• The product of two functions like $f\left(x\right)·g\left(x\right)$ is two functions multiplied together

• A composite function like $f\left(g\left(x\right)\right)$ is a function of a function

• For how to differentiate composite functions, see the "Chain rule" study guide

You do not need to know the proof of the formula for the product rule, as it is beyond the scope of the course.

The functions $f$ and $g$ are differentiable. The table shown gives values of the functions and their first derivatives at $x=2$.

Let $h$ be the function defined by $h\left(x\right)=f\left(x\right)·g\left(x\right)$. Find $h\text{'}\left(2\right)$. Show the work that leads to your answer.

Answer:

Apply the product rule to the function $h$

$h\text{'}\left(x\right)=f\text{'}\left(x\right)·g\left(x\right)+f\left(x\right)·g\text{'}\left(x\right)$

Substitute $x=2$

$h\text{'}\left(2\right)=f\text{'}\left(2\right)·g\left(2\right)+f\left(2\right)·g\text{'}\left(2\right)$

Use the values in the table

$h\text{'}\left(2\right)=\left(-1\right)·5+3·4$

$h\text{'}\left(2\right)=7$

Find the derivative of the following functions.

(a) $f\left(x\right)=e^{2x}\left(x^5+3x\right)$

(b) $g\left(x\right)=\ln 3x·\sin 2x$

Answer:

(a)

Assign $u$ and $v$ to each function

$u=e^{2x}$

$v=x^5+3x$

Find the derivatives of $u$ and $v$

$u^{\text{'}}=2e^{2x}$

$v^{\text{'}}=5x^4+3$

Apply the product rule, $y^{\text{'}}=u^{\text{'}}v+u{v}^{\text{'}}$

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

If you have used a different notation such as $y^{\text{'}}$ when working, you should write your final answer in a format that matches how the question was asked

$f^{\text{'}}\left(x\right)=2e^{2x}\left(x^5+3x\right)+e^{2x}\left(5x^4+3\right)$

This answer can also be factored

$f^{\text{'}}\left(x\right)=e^{2x}\left(2x^5+5x^4+6x+3\right)$

(b)

Assign $u$ and $v$ to each function

$u=\ln 3x$

$v=\sin 2x$

Find the derivatives of $u$ and $v$

$u^{\text{'}}=\frac{1}{x}$

$v^{\text{'}}=2\cos 2x$

Apply the product rule, $y^{\text{'}}=u^{\text{'}}v+u{v}^{\text{'}}$

$y^{\text{'}}=\frac{1}{x}·\sin 2x+\ln 3x·2\cos 2x$

Simplify

$g^{\text{'}}\left(x\right)=\frac{\sin 2x}{x}+2\ln \left(3x\right)\cos \left(2x\right)$