Math Core

You can expand $(x + 3)(x - 5)$ before differentiating, but you can't expand $x^2 \sin x$ or $e^x \ln x$ into simpler terms. To differentiate a product of two functions directly, you need the product rule. It is not the rule most people guess first.

The derivative of a product is not the product of derivatives

Try the tempting shortcut on a simple case. The function $x \cdot x = x^2$ has derivative $2x$ . Multiplying the derivatives of the factors gives $1 \cdot 1 = 1$ , which is wrong. The correct rule has two terms.

The product rule

If $f$ and $g$ are differentiable, then

\frac{d}{dx}\big[f(x)\,g(x)\big] = f'(x)\,g(x) + f(x)\,g'(x)

In words: the derivative of the first times the second, plus the first times the derivative of the second.

Check it on $x \cdot x$ : $1 \cdot x + x \cdot 1 = 2x$ . Correct.

Why it works

Picture the product $f(x)\,g(x)$ as the area of a rectangle with width $f(x)$ and height $g(x)$ . When $x$ increases by $h$ , the width grows by $\Delta f$ and the height grows by $\Delta g$ . The new area is the old rectangle plus three strips:

\Delta(\text{area}) = \Delta f \cdot g(x) + f(x) \cdot \Delta g + \Delta f \cdot \Delta g

Divide by $h$ and let $h \to 0$ . The first strip gives $f'(x)\,g(x)$ and the second gives $f(x)\,g'(x)$ . The tiny corner piece $\dfrac{\Delta f \cdot \Delta g}{h} = \dfrac{\Delta f}{h} \cdot \Delta g$ approaches $f'(x) \cdot 0 = 0$ , because $g$ is continuous (it's differentiable). What survives is exactly the product rule.

Using the rule

A reliable method: name the two factors, find each derivative, then assemble.

Worked example: A power times a trig function

Differentiate $y = x^2 \sin x$ .

Let $f = x^2$ and $g = \sin x$ . Then $f' = 2x$ and $g' = \cos x$ .

\frac{dy}{dx} = f'g + fg' = 2x\sin x + x^2 \cos x

Worked example: Exponential times logarithm

Differentiate $y = e^x \ln x$ .

With $f = e^x$ and $g = \ln x$ , you have $f' = e^x$ and $g' = \dfrac{1}{x}$ :

\frac{dy}{dx} = e^x \ln x + e^x \cdot \frac{1}{x} = e^x\left(\ln x + \frac{1}{x}\right)

Factoring out $e^x$ is optional, but it makes the expression easier to set equal to zero or evaluate later.

Should you simplify the result? On the AP exam, an unsimplified derivative like $2x\sin x + x^2\cos x$ earns full credit. Simplify only when it helps with what comes next, such as solving $f'(x) = 0$ , where factoring is usually the key step.

Products from a table

AP questions often give only values of $f$ , $g$ and their derivatives at a few points. You don't need formulas: plug the numbers straight into the product rule.

Worked example: Using a table of values

The table gives values of differentiable functions $f$ and $g$ .

$x$	$f(x)$	$f'(x)$	$g(x)$	$g'(x)$
2	5	$-1$	3	4

If $h(x) = f(x)\,g(x)$ , find $h'(2)$ .

h'(2) = f'(2)\,g(2) + f(2)\,g'(2) = (-1)(3) + (5)(4) = -3 + 20 = 17

Worked example: A tangent line

Find the tangent line to $y = xe^x$ at $x = 1$ .

The point is $(1, e)$ . By the product rule, $y' = 1 \cdot e^x + x e^x = e^x(1 + x)$ , so the slope at $x = 1$ is $2e$ . The tangent line is

y - e = 2e(x - 1), \quad \text{or} \quad y = 2ex - e

The tangent line to y = xeˣ at (1, e) has slope 2e ≈ 5.44.Open in grapher →

Common mistake

$\dfrac{d}{dx}\big[f(x)\,g(x)\big] \ne f'(x)\,g'(x)$ . The product rule always has two terms, and each term contains exactly one derivative. If your answer has a term with no derivative, or a term with two, look again.

Tip

A constant factor doesn't need the product rule: $\dfrac{d}{dx}[5\sin x] = 5\cos x$ by the constant multiple rule. (The product rule would also work; the constant's derivative is 0, so one term drops out.) Save the product rule for two factors that both contain $x$ .

Practice

Practice 1

Find $f'(x)$ for $f(x) = x^3 e^x$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 2

Find $\dfrac{dy}{dx}$ for $y = x\cos x$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 3

The functions $f$ and $g$ are differentiable, with $f(1) = 3$ , $f'(1) = -2$ , $g(1) = 4$ and $g'(1) = 5$ . If $h(x) = f(x)\,g(x)$ , find $h'(1)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

Let $f(x) = (x^2 + 1)(3x - 2)$ . Use the product rule to find $f'(1)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Let $f(x) = x \ln x$ . Find $f'(e)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

What is $\dfrac{d}{dx}\big[\sin x \cos x\big]$ ?

Practice 7

Find all $x$ in the interval $0 \le x \lt 2\pi$ where $f(x) = e^x \sin x$ has a horizontal tangent line.

Separate answers with commas, e.g. 2, -5

Practice 8

Find an equation of the tangent line to $y = xe^x$ at $x = 0$ .

Enter an expression, e.g. 3x^2 - 2x + 1