the Product Rule?

The formula for differentiating two functions multiplied together — it's not simply $f' \cdot g'$, but $f'g + fg'$.

Simple Definition

When two functions are multiplied together, their combined rate of change is not just the product of their individual rates. The product rule gives the correct formula:

$(fg)' = f'g + fg'$

Read it as: "derivative of first times second, plus first times derivative of second." This rule is unavoidable whenever you differentiate expressions like $x^2 \sin x$, $e^x \ln x$, or $(x^3+1)(x^2-2)$.

Intuition — The Rectangle Argument

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Area of a changing rectangle

Imagine a rectangle with width $f(x)$ and height $g(x)$. Its area is $A = f \cdot g$. When $x$ increases by a tiny $\Delta x$, both $f$ and $g$ change. The new area has three extra pieces: a strip of area $\Delta f \cdot g$ (top), a strip of area $f \cdot \Delta g$ (right), and a tiny corner $\Delta f \cdot \Delta g$ (negligible). Dividing by $\Delta x$ and taking the limit gives $f'g + fg'$. The product rule is literally the calculus of a growing rectangle.

The two strips represent f′g (left strip) and fg′ (top strip) — the product rule geometrically

Formal Statement

Product Rule

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

Leibniz Notation

$$\frac{d}{dx}[uv] = \frac{du}{dx}\cdot v + u\cdot\frac{dv}{dx}$$

How to Apply It — Step by Step

Differentiate $h(x) = x^2 \cdot \sin x$:

Identify $f$ and $g$

$f(x) = x^2$, \quad $g(x) = \sin x$

Differentiate each

$f'(x) = 2x$, \quad $g'(x) = \cos x$

Apply the formula

$h'(x) = f'g + fg' = 2x \cdot \sin x + x^2 \cdot \cos x$

Quick Examples

Example 1

$x^3 \cdot e^x$

$f=x^3,\ g=e^x$
$= 3x^2 e^x + x^3 e^x$
$= e^x(3x^2+x^3)$

Example 2

$\ln x \cdot \cos x$

$f=\ln x,\ g=\cos x$
$= \dfrac{1}{x}\cos x + \ln x(-\sin x)$
$= \dfrac{\cos x}{x} - \ln x \sin x$

Example 3 — Triple product

$x \cdot e^x \cdot \sin x$

Apply twice: $(fg)h$
$= e^x\sin x + xe^x\sin x + xe^x\cos x$

Example 4 — Could expand

$(x+1)(x^2-3)$

Product rule: $1\cdot(x^2-3)+(x+1)(2x)$
$= x^2-3+2x^2+2x = 3x^2+2x-3$

Common Mistakes

✗ (fg)′ = f′ · g′

This is one of the most common errors in calculus. The derivative of a product is NOT the product of the derivatives. $(x^2 \cdot e^x)' \neq 2x \cdot e^x$. You must use $f'g + fg'$.

✗ Applying the power rule to a product

$(x \cdot \sin x)' \neq (x \cdot \sin x)^0 \cdot 1$. The power rule only applies to a single variable raised to a power, not to a product. Always identify the structure first.

✗ Writing only one term

The product rule always gives TWO terms: $f'g$ AND $fg'$. Writing only $f'g$ without the second term $fg'$ is incomplete and wrong. Both are essential.

Related Concepts

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The product rule is tested extensively in Calculus I on BUders.

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