Topic 2.8 Notes – The Product Rule

The Product Rule

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

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

Memory phrase you’ll hear in class:

“First d second + second d first.”

That phrase is worth memorizing exactly. It keeps you from dropping a term.

Very important:

$$(fg)' \ne f'g'$$

If you multiply the derivatives, you are missing half the answer.

Why You Need It

You already know:

• Power rule
• Derivatives of $\sin x, \cos x$
• Derivatives of $e^x, \ln x$

Those rules work when a function stands alone.

But when you see something like
$$x^3\cos x$$ both pieces change as $x$ changes. Expanding isn’t possible. That’s when the Product Rule is required.

Think of it this way: when two changing quantities multiply, both changes affect the result.

When to Use the Product Rule

Use it when:

• Two non-constant expressions are multiplied
  • $x^4 \ln x$
  • $(2x-1)\sin x$
  • $e^x(5x^2+3)$

Do NOT use it when:

• One factor is just a constant
  • $7x^5$ → constant multiple rule
• The expression is already a single polynomial
  • $6x^3 - 2x$ → just use power rule

If it’s polynomial × polynomial, you can either:

• Expand first
• Or use Product Rule

Both are valid. On a no-calculator MCQ, expanding is sometimes faster.

How It Actually Works

Let’s walk through one clean example.

Find the derivative of
$$y = (x^2 + 1)e^x$$

Step 1: Identify the factors

• $f(x) = x^2 + 1$
• $g(x) = e^x$

Step 2: Differentiate each

• $f'(x) = 2x$
• $g'(x) = e^x$

Step 3: Apply the formula

$$y' = (x^2 + 1)(e^x) + (e^x)(2x)$$

That is already a complete, correct answer.

You could factor:

$$y' = e^x(x^2 + 1 + 2x)$$

But unless the problem says “simplify,” the unsimplified form earns full credit on FRQs.

What the Structure Looks Like

Every correct product rule derivative has two terms, and each term contains one original factor.

Here’s the standard structure you should recognize instantly:

Product Rule structure: differentiate one factor at a time

Notice in the formula above:

• The first term keeps $v(x)$ and differentiates $u(x)$
• The second term keeps $u(x)$ and differentiates $v(x)$

If one original function completely disappears, something went wrong.

Common Product Types on Tests

Polynomial × Trig

Example structure: $x^3\sin x$

• Differentiate trig carefully:
  • $(\sin x)' = \cos x$
  • $(\cos x)' = -\sin x$

Sign mistakes here are very common on quizzes.

Polynomial × Exponential

Example: $x^2 e^x$

Remember:

$$(e^x)' = e^x$$

Students sometimes forget that $e^x$ stays the same.

Exponential × Trig

Example: $e^x \cos x$

You’ll often factor afterward:

$$e^x\cos x - e^x\sin x = e^x(\cos x - \sin x)$$

These show up in differential equation contexts later in BC.

Mistakes That Cost Easy Points

• Writing only one term
• Multiplying derivatives together
• Forgetting parentheses when differentiating a binomial
• Dropping a negative from $(\cos x)' = -\sin x$

On FRQs, missing the second term usually loses most of the derivative points immediately.

How It Gets Tested

You’ll see it:

• On no-calculator multiple choice mixed with trig and exponentials
• Embedded inside larger FRQs (motion, accumulation, implicit differentiation)
• Combined with the chain rule, like $(x^2+1)\sin(3x)$

When rules combine, apply Product Rule first, then Chain Rule inside the pieces that need it.