Derivative Rules: Power, Product, Quotient and Chain Rule

Derivative rules are shortcuts for finding the derivative of a function without using the limit definition every time. The core rules are the power rule (xⁿ becomes n·xⁿ⁻¹), the product rule (f·g becomes f′g + fg′), the quotient rule (f/g becomes (f′g − fg′) ÷ g²) and the chain rule (f(g(x)) becomes f′(g(x))·g′(x)). Combined with the derivatives of common functions like sin x, eˣ and ln x, they cover nearly every derivative in an introductory calculus course.

What a derivative tells you

The derivative f′(x) is the instantaneous rate of change of f at x, which is also the slope of the tangent line to the graph at that point. It's defined by a limit:

f′(x) = lim (h → 0) of [f(x + h) − f(x)] ÷ h

Every rule below can be proved from this definition, but once you have the rules, you rarely need the limit. Notation varies: f′(x), dy/dx and d/dx [f(x)] all mean the same thing.

The basic rules

Power rule

Bring the exponent down as a coefficient, then subtract one from the exponent. It works for any real exponent, including negatives and fractions.

Example 1. f(x) = 4x³ − 5x² + 7x − 2

f′(x) = 12x² − 10x + 7

The constant −2 disappears, and 7x becomes 7 because x = x¹ and 1·x⁰ = 1.

Example 2. f(x) = 3/x² + 2√x

Rewrite with exponents first: f(x) = 3x⁻² + 2x^(1/2).

f′(x) = −6x⁻³ + x^(−1/2) = −6/x³ + 1/√x

Rewriting roots and fractions as powers before differentiating is the single most useful habit for the power rule.

Product rule

When two functions are multiplied, differentiate each one in turn while leaving the other alone, then add:

(f·g)′ = f′·g + f·g′

Example 3. y = x²·sin x

Let f = x² (f′ = 2x) and g = sin x (g′ = cos x).

y′ = 2x·sin x + x²·cos x

Quotient rule

For a fraction f/g:

(f/g)′ = (f′·g − f·g′) ÷ g²

The order matters because of the minus sign. A common memory aid is "low d-high minus high d-low, over low squared."

Example 4. y = (x² + 1) ÷ (x − 3)

Let f = x² + 1 (f′ = 2x) and g = x − 3 (g′ = 1).

y′ = [2x(x − 3) − (x² + 1)(1)] ÷ (x − 3)²
   = (2x² − 6x − x² − 1) ÷ (x − 3)²
   = (x² − 6x − 1) ÷ (x − 3)²

Chain rule

Use the chain rule for a function inside another function. Differentiate the outer function, keep the inner function as it is, then multiply by the derivative of the inner function.

d/dx [f(g(x))] = f′(g(x))·g′(x)

Example 5. y = (3x² + 1)⁵

Outer: u⁵ with u = 3x² + 1. Inner derivative: 6x.

y′ = 5(3x² + 1)⁴·6x = 30x(3x² + 1)⁴

Example 6. A few quick chain rule cases:

• d/dx of sin(4x) = cos(4x)·4 = 4cos(4x)
• d/dx of e^(x²) = e^(x²)·2x = 2x·e^(x²)
• d/dx of ln(x² + 1) = (1 ÷ (x² + 1))·2x = 2x ÷ (x² + 1)
• d/dx of √(5x + 2) = ½(5x + 2)^(−1/2)·5 = 5 ÷ (2√(5x + 2))
• d/dx of sin²x = 2 sin x·cos x

Forgetting to multiply by the inner derivative is the most common chain rule error. If the inside is anything other than plain x, the chain rule applies.

Derivatives of common functions

The trigonometric derivatives assume x is measured in radians.

Combining rules

Real problems often need more than one rule. Work from the outside in: decide what the main operation is (a product, a quotient, a composition), apply that rule, and use other rules for the pieces.

Example 7. y = x³·e^(2x)

This is a product. The second factor needs the chain rule: its derivative is 2·e^(2x).

y′ = 3x²·e^(2x) + x³·2e^(2x) = x²·e^(2x)·(3 + 2x)

Factoring the answer isn't required, but it makes the result easier to use, for instance when you set y′ = 0.

Using derivatives: slopes and tangent lines

Example 8. Find the tangent line to f(x) = x³ − 2x at x = 2.

1. Point: f(2) = 8 − 4 = 4, so the point is (2, 4).
2. Derivative: f′(x) = 3x² − 2
3. Slope: f′(2) = 12 − 2 = 10
4. Line: y − 4 = 10(x − 2), so y = 10x − 16

Check: at x = 2, the line gives 20 − 16 = 4, matching the curve ✓.

Setting the derivative to zero finds where a curve is flat. For the parabola y = x² − 4x − 1, y′ = 2x − 4 = 0 gives x = 2, so the vertex is at (2, −5). That's the same parabola solved by completing the square in our guide on how to solve quadratic equations.

Higher derivatives

Differentiate again to get the second derivative f″(x), which describes how the slope is changing (concavity, or acceleration in physics).

For f(x) = x⁴: f′(x) = 4x³, f″(x) = 12x², f‴(x) = 24x.

Common mistakes

• Product as product of derivatives. (f·g)′ is not f′·g′.
• Quotient rule order. f′g − fg′, not fg′ − f′g. Swapping them flips the sign.
• Missing the inner derivative in the chain rule.
• Treating constants as variables. The derivative of π² or e³ is 0, since they're just numbers.
• Not rewriting first. 1/x³ and ∛x are much easier as x⁻³ and x^(1/3).
• Degrees instead of radians for trig derivatives.

Checking your work

Differentiate by hand first, then compare with a tool. A derivative calculator that names the rule used for each term helps you see exactly where your answer split from the correct one. For word problems or proofs, a step-by-step solver can explain the reasoning, but treat AI explanations as a study aid, check them, and follow your course's rules on tools; our guide on using AI without violating academic integrity explains the usual boundaries.

Derivative CalculatorDerivatives with the rule for each step

Derivatives describe change at a single instant. For change between two points, such as a price over a year, see how to calculate percentage change.

Key takeaways

• Power rule: xⁿ → n·xⁿ⁻¹. Rewrite roots and fractions as powers first.
• Product rule: (fg)′ = f′g + fg′. Quotient rule: (f/g)′ = (f′g − fg′) ÷ g².
• Chain rule: differentiate the outside, keep the inside, multiply by the inside's derivative.
• Memorize the short table of common derivatives, use radians for trig, and check answers at a point.