Derivative Rules — Power, Product, Quotient & Chain

The hardest part of most derivative problems is not the algebra; it is choosing the right rule. Derivative rules tell you which formula fits the structure of a function, and the single habit that makes them work is reading the outside structure first, then differentiating from there.

When To Use Each Rule

Before any algebra, ask what the outermost structure of the expression is, then match a rule to it:

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

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

The Rules Themselves

Power rule

If $n$ is a real constant,

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

For example, $\frac{d}{dx}(x^5) = 5x^4$. Use this for 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,

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

The derivative has two terms because either factor can cause the product to change.

Quotient rule

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

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

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,

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

In plain language: differentiate the outer function, keep the inner expression in place, then multiply by the derivative of the inner expression.

A Worked Pass: Product 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 $f(x) = x^2$ and $g(x) = (3x+1)^4$. Then

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

Differentiate the first factor:

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

Differentiate the second 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 correct. For a cleaner factored form, pull out the common pieces:

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

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

Where The Steps Break, And How To Verify

Picking the rule. Applying the power rule to the whole expression when the function is really a product or quotient is the classic misfire. Self-check: name the outermost structure out loud before writing anything.

Building the product term. Writing the derivative of a product as $f'(x)g'(x)$ instead of two added terms drops half the answer. Self-check: a product derivative always has two terms.

The quotient numerator. Forgetting the minus sign changes the result entirely. Self-check: the term with $f'$ comes first and is positive.

The chain-rule inner piece. Turning $(3x+1)^4$ into just $4(3x+1)^3$ omits the inner derivative. Self-check: every composite leaves an inner-derivative factor behind.

Expanding too early. Multiplying out first often makes the algebra harder. Self-check: if the structure is clear, apply the rule directly.

Where These Rules Are Used

Derivative rules matter anywhere you need a rate of change: tangent slopes, motion, optimization, and graph behavior in a calculus course; velocity and acceleration in physics; and how one quantity responds when another changes in engineering or economics.

Try The Structure Check

Differentiate

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

This is a good test because the outside form is a quotient, while the denominator also needs the chain rule. For a close comparison, explore the Chain Rule or Product Rule next.