Derivatives for Beginners: Power, Product and Chain Rules

A derivative tells you the slope of a function's graph at every point: how fast the output changes as the input changes. You find it with a handful of rules. The power rule handles $x^n$, the product and quotient rules handle multiplication and division, and the chain rule handles one function inside another.

What a derivative means

The slope of a straight line is rise over run. A curve has a different slope at every point, and the derivative $f'(x)$ gives that slope. If $f(x)$ is a position, $f'(x)$ is a speed.

Rule 1: The power rule

Bring the power down in front, then reduce the power by one. Constants differentiate to $0$, and constant multiples stay in front.

Example: $f(x) = x^3 - 4x + 1$

At $x = 2$ the slope is $3(4) - 4 = 8$. The tangent line there rises 8 units for every 1 unit across.

Rule 2: The product rule

When two functions are multiplied:

Example: $f(x) = x^2 \sin x$, with $u = x^2$ and $v = \sin x$:

Rule 3: The quotient rule

Rule 4: The chain rule

For a function inside another function:

Differentiate the outside, keep the inside the same, then multiply by the derivative of the inside.

Example: $\sin(2x)$. The outside is $\sin$, the inside is $2x$:

Standard derivatives to learn

Common mistakes

• Forgetting the chain rule. $\frac{d}{dx}(3x+1)^5$ is $15(3x+1)^4$, not $5(3x+1)^4$.
• Treating a product as two separate derivatives. $(x^2\sin x)' \ne 2x\cos x$.
• Sign of $\cos$. The derivative of $\cos x$ is $-\sin x$.

Practice questions

1. Differentiate $5x^4 - 2x^2 + 7$.
2. Differentiate $x e^x$.
3. Differentiate $(x^2 + 1)^3$.

Answers: 1) $20x^3 - 4x$ 2) $e^x + xe^x$ 3) $6x(x^2 + 1)^2$

Type any function into the derivative calculator to see every rule it uses, along with the tangent line. You can compare $f$ and $f'$ in the graphing calculator. Next, see how integration reverses differentiation.

Frequently asked questions

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