Chain Rule in Calculus: Formula & Examples

Definitionnoun phrase

What is the chain rule in calculus?

The chain rule is the rule for the derivative of a composite function: if y = f(g(x)), then y′ = f′(g(x)) · g′(x). Differentiate the outer function, leave the inner function inside, then multiply by the derivative of the inner function. For (x² + 1)³, the derivative is 3(x² + 1)² · 2x.

In Spanish: regla de la cadena
Also called: chain rule for derivatives, composite function rule, outside-inside rule
Grade level: Calculus

CalculusAP

Chain Rule in a picture

Peel the function from the outside in: the outer power gives 4(3x + 1)³, and the inner function contributes a factor of 3. Together, 12(3x + 1)³.

Formal definition and notation

If g is differentiable at x and f is differentiable at g(x), then the composite h(x) = f(g(x)) is differentiable at x and h′(x) = f′(g(x)) g′(x). In Leibniz notation, if y = f(u) and u = g(x), then dydx = dydu · dudx. For a chain of three functions the rule repeats: (f(g(k(x))))′ = f′(g(k(x))) g′(k(x)) k′(x).

Symbol	Read it as	Meaning
f(g(x))	f of g of x	a composite function: g inner, f outer
f′(g(x)) · g′(x)	f prime of g of x, times g prime of x	the chain rule
dydu · dudx	d y d u times d u d x	Leibniz form: rates of change multiply

Key facts

Spot the chain rule whenever something other than plain x sits inside a power, root, exponential, log or trig function.
The inner function stays the same in the first factor; only its derivative is multiplied on.
Power chain: ddx(5x − 2)⁶ = 6(5x − 2)⁵ · 5.
Exponential chain: ddxe^3x = 3e^3x. Log chain: ddxln (x² + 1) = 2xx² + 1.
If the inner function is just x, its derivative is 1 and the chain rule changes nothing.
Leibniz form explains the name: rates link like a chain. If u changes 3 times as fast as x and y changes 2 times as fast as u, then y changes 6 times as fast as x.

Chain rule

ddx( f(g(x)) ) = f′(g(x)) · g′(x)

g(x): the inner function
f′(g(x)): the derivative of the outer function, with the inner left unchanged inside
g′(x): the derivative of the inner function

Power form, the most common case: ddx( uⁿ ) = nu^{n − 1} · u′.

Chain Rule examples, step by step

Example 1: Power of a linear function

Warm-up

Differentiate y = (2x − 5)³.

Outer: u³, so its derivative is 3u² = 3(2x − 5)².
Inner: u = 2x − 5, so u′ = 2.
Multiply: 3(2x − 5)² · 2 = 6(2x − 5)².

Answer: y′ = 6(2x − 5)².

Example 2: A square root

Core

Find f′(x) for f(x) = √x² + 9, then evaluate f′(4).

Write f(x) = (x² + 9)¹².
Outer derivative: 12(x² + 9)⁻¹². Inner derivative: 2x.
Multiply: f′(x) = 2x2√x² + 9 = x√x² + 9.
f′(4) = 4√25 = 45.

Answer: f′(x) = x√x² + 9, so f'(4) = 4/5.

Example 3: Chain rule from a table

Test-style

Let h(x) = f(g(x)). You know g(2) = 5, g′(2) = 3, f(5) = 1 and f′(5) = −4. Find h′(2).

Chain rule: h′(2) = f′(g(2)) · g′(2).
Inner value first: g(2) = 5, so you need f′(5) = −4.
Multiply: −4 · 3 = −12.

Answer: h'(2) = -12

Common mistakes

✗ Mistakeddx(x² + 1)³ = 3(x² + 1)².

✓ Instead3(x² + 1)² · 2x = 6x(x² + 1)².

This forgets the inner derivative, the most common chain rule error. The inside is x² + 1, not x, so its derivative 2x must be multiplied on.

✗ Mistakeddx(x² + 1)³ = 3(2x)².

✓ Instead3(x² + 1)² · 2x.

The inner function stays unchanged inside the outer derivative. Its derivative goes outside as a separate factor.

✗ MistakeUsing f′(2) for h′(2) when h = f(g(x)).

✓ InsteadUse f′(g(2)).

The outer derivative is evaluated at the inner function’s output, not at the original input.

Not to be confused with

Chain Rulevs.Product rule

The product rule is for two functions multiplied, f(x) · g(x). The chain rule is for one function inside another, f(g(x)). x²e^x needs the product rule; e^x² needs the chain rule.

Chain Rulevs.Power rule

The power rule alone handles xⁿ. When the base is an expression, like (4x + 1)ⁿ, you need the power rule and the chain rule.

Quick check: 3 questions

Question 1 of 3

What is ddx(x² + 1)⁵?

Show answer and explanation

Answer: B. Outer: 5(x² + 1)⁴; inner derivative: 2x. The product is 10x(x² + 1)⁴. The first choice forgets the inner derivative and the third replaces the inside with its derivative.

Question 2 of 3

What is ddxe^4x?

Show answer and explanation

Answer: C. The derivative of e^u is e^u, times u′ = 4: 4e^4x. e^4x forgets the inner derivative, and 4xe^{4x − 1} wrongly uses the power rule on an exponential.

Question 3 of 3

Which function needs the chain rule?

Show answer and explanation

Answer: D. √5x − 1 has the inner function 5x − 1 inside a root. The others are powers of plain x, handled by the power rule alone.

Chain Rule: frequently asked questions

When do you use the chain rule?

Use it whenever you differentiate a function of a function: an expression other than plain x inside a power, root, exponential, logarithm or trig function. Examples: (3x − 7)⁴, √x² + 1, e^5x, ln (2x). If the inside is just x, as in x⁴, the chain rule is not needed.

What is the chain rule formula?

For y = f(g(x)), the derivative is f′(g(x)) · g′(x): the outer derivative with the inner function left inside, times the inner derivative. In Leibniz notation, dydx = dydu · dudx, where u is the inner function. The common power form is ddxuⁿ = nu^{n − 1}u′.

Why is it called the chain rule?

Because rates of change link together like a chain. If y depends on u, and u depends on x, then a small change in x changes u, which in turn changes y. The overall rate is the product of the links: dydu times dudx. With more nested functions, the chain just gets longer.

Practice and tools

Derivative CalculatorDifferentiate any function and see each rule used Tangent Line CalculatorTurn a derivative value into the equation of the tangent line

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: n/a

Cite this entry

Effortless Math Team. “Chain Rule.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/chain-rule/

Flashcard

TermChain Ruleddxf(g(x))

DefinitionOuter derivative with the inner left inside, times the inner derivative: f′(g(x))g′(x). Example: (3x + 1)⁴ → 12(3x + 1)³.

Study the full deck

Browse all terms in the Math Dictionary · More calculus terms · Random term

Composition of FunctionsA composition of functions applies one function to the output of another. The composite f(g(x)), also written (f ∘ g)(x), means: find g(x) first, then put that result into f. If g(x) = x + 1 and f(x) = x², then f(g(2)) = f(3) = 9. Order matters.Functions · Algebra 2, PrecalculusPowerA power is a number written as a base with an exponent, such as 5³, or the value that results, 5³ = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7DerivativeA derivative measures how fast a function’s output changes at one exact input: its instantaneous rate of change. On a graph it is the slope of the tangent line at that point. The derivative of f(x) = x² is f′(x) = 2x, so at x = 3 the curve is rising with slope 6.Calculus · CalculusEvaluateEvaluate means to find the single number value of an expression. Replace each variable with the number you are given, then calculate using the order of operations. For example, 2x + 7 evaluated at x = 3 is 2(3) + 7 = 13. The result is one number, not a new expression.Algebra · Grade 6, Grade 7, Algebra 1Power RuleThe power rule is the shortcut for the derivative of a power of x: if f(x) = xⁿ, then f′(x) = nx^{n − 1}. Bring the exponent down as a coefficient and subtract 1 from it, so x⁵ becomes 5x⁴. It works for any real exponent, including negative and fractional ones.Calculus · CalculusLogarithmA logarithm is the exponent you need to raise a base to in order to get a given number. log_b x = y means b^y = x, so log₂ 32 = 5 because 2⁵ = 32. A log with no base written uses base 10: log 1000 = 3. Logs undo exponents.Functions · Algebra 2, PrecalculusFunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2Critical PointA critical point of a function is an input x = c in its domain where the derivative is 0 or does not exist. These are the only places a smooth graph can turn around, so every local maximum or minimum happens at a critical point. For f(x) = x² − 4x, f′(x) = 2x − 4 = 0 at x = 2.Calculus · CalculusIntegralAn integral adds up infinitely many tiny pieces to find a total, such as the area under a curve. The definite integral ∫^b_a f(x) dx is the signed area between the graph of f and the x-axis from x = a to x = b. An indefinite integral is an antiderivative: ∫ 2x dx = x² + C.Calculus · CalculusEuler's NumberEuler’s number, written e, is an irrational constant approximately equal to 2.71828. It is the number that ( 1 + 1n )ⁿ gets closer and closer to as n grows. It is the base of natural exponential growth, e^x, and of the natural logarithm, ln x.Functions · Algebra 2, Precalculus, CalculusNatural LogarithmThe natural logarithm of a positive number x, written ln x, is the power you must raise e ≈ 2.718 to in order to get x. So ln x = y means e^y = x. For example, ln 1 = 0, ln e = 1, and ln 20 ≈ 3.0 because e³ ≈ 20.1.Functions · Algebra 2, Precalculus, CalculusRate of ChangeA rate of change tells how much an output changes for each 1-unit change in the input: the change in y divided by the change in x. If a phone plan costs 15 dollars more for every extra hour, its rate of change is 15 dollars per hour. For a straight line, the rate of change equals the slope.Functions · Grade 8, Algebra 1, Algebra 2

Chain Rule

What is the chain rule in calculus?

Chain Rule in a picture

Formal definition and notation

Key facts

Chain rule

Chain Rule examples, step by step

Example 1: Power of a linear function

Example 2: A square root

Example 3: Chain rule from a table

Common mistakes

Not to be confused with

Quick check: 3 questions

Chain Rule: frequently asked questions

When do you use the chain rule?

What is the chain rule formula?

Why is it called the chain rule?

Related terms

Practice and tools

Sources

Flashcard

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