Derivative Rules — Power, Product, Quotient & Chain

Derivative rules tell you which differentiation formula fits the structure of a function. If the expression is a power, product, quotient, or nested function, choose the rule for that outside structure first. That one habit makes most derivative problems much easier.

The main derivative rules and when to use them

Power rule

If $n$ is a real constant, then

$$\frac{d}{dx}(x^n) = nx^{n-1}$$

Example: $\frac{d}{dx}(x^5) = 5x^4$.

Use this when the expression is a plain power of $x$. If the base is not just $x$, such as $(3x+1)^5$, the chain rule is also involved.

Product rule

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

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

Use this when two changing expressions are multiplied. The derivative has two terms because either factor can cause the product to change.

Quotient rule

If $f$ and $g$ are differentiable and $g(x) \ne 0$, then

$$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$$

Use this when one changing expression is divided by another. The condition $g(x) \ne 0$ matters because the original function is undefined where the denominator is zero.

Chain rule

If $y = f(g(x))$, and both functions are differentiable where needed, then

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

Use this when one function is inside another. In plain language: differentiate the outer function, keep the inner expression in place, then multiply by the derivative of the inner expression.

How to tell which derivative rule to use

Do not start by hunting for a memorized formula. Start by asking: what is the outermost structure of the expression?

• $x^7$ is a power.
• $x^2\sin(x)$ is a product.
• $\frac{x^2+1}{x-3}$ is a quotient.
• $(2x-1)^4$ or $\sin(x^2)$ is a composite function, so the chain rule applies.

If an expression mixes structures, start with the outside one. For example, $x(2x-1)^4$ is a product overall, even though one factor also needs the chain rule.

Worked example: product rule with a chain rule inside

Find the derivative of

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

The outside structure is a product, so use the product rule first. Let

$$f(x) = x^2 \quad \text{and} \quad g(x) = (3x+1)^4$$

Then

$$y' = f'(x)g(x) + f(x)g'(x)$$

Differentiate the first factor:

$$f'(x) = 2x$$

Differentiate the second factor with the chain rule:

$$g'(x) = 4(3x+1)^3 \cdot 3 = 12(3x+1)^3$$

Substitute both parts:

$$y' = 2x(3x+1)^4 + x^2 \cdot 12(3x+1)^3$$

This is already a correct final answer. If you want a cleaner factored form, pull out the common pieces:

$$y' = 2x(3x+1)^3(9x+1)$$

The key idea is the order. Choose the product rule from the outside structure, then use the chain rule only where it is needed inside the factor $(3x+1)^4$.

Common mistakes with derivative rules

1. Using the power rule on the whole expression when the function is actually a product or quotient.
2. Writing the derivative of a product as $f'(x)g'(x)$ instead of two added terms.
3. Forgetting the minus sign in the quotient rule numerator.
4. Forgetting the inner derivative in the chain rule, as in turning $(3x+1)^4$ into just $4(3x+1)^3$.
5. Expanding early and making the algebra harder than it needs to be.

Where these rules are used in calculus

Derivative rules matter anywhere you need a rate of change. In a calculus course, that usually means tangent slopes, motion, optimization, and graph behavior. In physics, they show up in velocity and acceleration. In engineering or economics, they help describe how one quantity responds when another changes.

Try a similar problem

Differentiate

$$y = \frac{x^2+1}{(2x-3)^2}$$

This is a good structure check because the outside form is a quotient, while the denominator also needs the chain rule.

If you want another close comparison, explore the Chain Rule or Product Rule next.