2.8 The Product Rule

The product rule lets you find the derivative of two functions multiplied together. The formula is $\frac{d}{dx}(f(x)g(x)) = f(x)g'(x) + g(x)f'(x)$, which you can remember as "first times derivative of second, plus second times derivative of first." You cannot just multiply the two derivatives together, which is the most common mistake. For AP Calculus, write both terms before simplifying.

Why This Matters for the AP Calculus Exam

The product rule is one of the core differentiation tools you need for AP Calculus. Once functions get more complicated than simple polynomials, you will run into products like $x^2 \sin x$ or $e^x \cos x$ that cannot be handled by the power rule or sum rule alone.

You can expect product rule questions in both multiple-choice and free-response sections. Free-response questions sometimes give you a table of values for two functions and ask you to find the derivative of their product at a specific point. In that case, showing the product rule structure clearly is important for earning full credit, even when the final number is correct. The product rule also sets up the quotient rule and combines with the chain rule in later units, so getting comfortable with it now pays off across the whole course.

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Key Takeaways

• The product rule formula is $\frac{d}{dx}(f(x)g(x)) = f(x)g'(x) + g(x)f'(x)$.
• The derivative of a product is not the product of the derivatives.
• Differentiate both factors, but only one factor at a time in each term, then add the results.
• Both functions must be differentiable for the product rule to apply.
• On the AP exam you usually do not need to simplify your answer unless the problem asks for it.
• When a question gives values in a table, write out the full product rule structure like $h'(3) = f(3)g'(3) + g(3)f'(3)$ for $h(x)=f(x)g(x)$ before plugging in numbers.

The Product Rule

To differentiate a product of functions, multiply the first function by the derivative of the second and add the second function times the derivative of the first.

$\frac{d}{dx}(\textcolor{red}{f(x)}\textcolor{green}{g(x)})= \textcolor{red}{f(x)} \textcolor{blue}{g'(x)} + \textcolor{green}{g(x)} \textcolor{pink}{f'(x)}$

A simple way to remember it:

"First d second" (first function times the derivative of the second)

Plus "second d first" (second function times the derivative of the first).

This rule is necessary because the product of the derivatives of two functions does not equal the derivative of their product.

Product Rule Walkthrough

Find the derivative of:

$f(x) = \sin(x)(x^2 + 2x)$

Using the product rule:

$\textcolor{green}{f'(x)} = \sin(x)*\frac {d}{dx}(x^2 + 2x) + (x^2+2x)*\frac {d}{dx}(\sin(x))$

$\textcolor{green}{f'(x)} = \sin(x)(2x+2) + (x^2+2x)\cos(x)$

If you incorrectly tried to multiply the derivatives, you would get:

$\textcolor{red}{f'(x)} = \cos(x)( 2x + 2)$

But $\textcolor{green}{\sin(x)(2x+2) + (x^2+2x)\cos(x)} \neq \textcolor{red}{\cos(x)( 2x + 2)}$.

The graphs below show why. The correct $f'(x)$ matches the original function because its critical points and its positive and negative regions line up with where $f(x)$ rises and falls.

How to Use This on the AP Calculus Exam

Problem Solving

Work through these to build fluency with the product rule.

Example 1. Find $y'$ for $y = (3x^2-4x)(2x-1)$ both with and without the product rule.

Without the product rule: expand first.

$y = 6x^3 - 3x^2- 8x^2 + 4x$

$y = 6x^3 -11x^2 + 4x$

Now take the derivative using the sum and difference rules:

$y' = 18x^2 - 22x + 4$

With the product rule:

$y' = (3x^2-4x)* \frac{d}{dx}(2x-1) + (2x-1)* \frac{d}{dx}(3x^2-4x)$

$y' = (3x^2-4x)(2) + (2x-1)(6x-4)$

Both methods give the same answer once expanded. Unless a problem says to simplify, this form is perfectly acceptable.

Example 2. Find $f'(x)$ if $f(x) = \sin(x)(3x^2 - 2x + 5)$.

Remember, the derivative of $\sin(x)$ is $\cos(x)$. For a refresher, see Derivatives of cos x, sin x, e^x, and ln x.

$f'(x) = \sin(x)*\frac{d}{dx}(3x^2-2x+5) + (3x^2-2x+5)*\frac{d}{dx}\sin(x)$

$f'(x) = \sin(x)(6x-2) + (3x^2-2x+5)\cos(x)$

Example 3. Find $y'$ if $y = e^x\sin(x)$.

The derivative of $e^x$ is still $e^x$, so:

$y'=e^x * \frac{d}{dx}(\sin(x)) + \sin(x)*\frac{d}{dx}(e^x)$

$y' = e^x\cos(x) + e^x\sin(x)$

Common Trap

When a free-response question gives a table of values for $u$ and $v$ and asks for the derivative of $u(x)v(x)$ at a point, write the product rule structure first. For example, $f'(3) = u(3)v'(3) + v(3)u'(3)$, then substitute the values from the table. Showing this structure is important for clear exam work, even if your final number is right.

Common Misconceptions

• The derivative of a product is not the product of the derivatives. $\frac{d}{dx}(fg) \neq f'g'$. This is the single most common product rule error.
• You still differentiate both factors. Some students only differentiate one function. Each term in the result keeps one factor undifferentiated and differentiates the other.
• Order of terms does not matter for adding. Since you are adding two products, $f g' + g f'$ and $g f' + f g'$ are the same. Just make sure each term pairs an original function with the derivative of the other.
• Expanding is an option, not a requirement. For simple polynomial products you can multiply out and use the power rule, but for products like $e^x \sin x$ the product rule is the practical choice.
• You usually do not need to simplify. A correct unsimplified answer earns credit unless the problem specifically asks for a simplified form.

Related AP Calculus Guides

• Unit 2 Overview: Differentiation
• 2.1 Defining Average and Instantaneous Rates of Change at a Point
• 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
• 2.2 Defining the Derivative of a Function and Using Derivative Notation
• 2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
• 2.3 Estimating Derivatives of a Function at a Point