What is the chain rule in calculus?

The chain rule is a formula for finding the derivative of a composite function. If you have f(g(x)), the chain rule says the derivative is f'(g(x)) · g'(x). You differentiate the outer function, keep the inner function unchanged, then multiply by the derivative of the inner function.

How do I know when to use the chain rule?

Use the chain rule whenever you're differentiating a "function within a function" — any time the argument of a function is more complex than just the variable x. For example, sin(x²), e^(3x), (2x+1)^5, and ln(cos(x)) all require the chain rule because their arguments are expressions, not just x.

What is the chain rule formula in Leibniz notation?

In Leibniz notation, if y = f(u) and u = g(x), the chain rule is dy/dx = (dy/du) · (du/dx). This notation is intuitive because the du terms appear to "cancel" like fractions, leaving dy/dx. It's especially useful when working with related rates and implicit differentiation.

How do I apply the chain rule step by step?

Follow these steps: (1) Identify the composite function f(g(x)). (2) Determine the outer function f(u) and inner function u = g(x). (3) Differentiate the outer function to get f'(u). (4) Differentiate the inner function to get g'(x). (5) Multiply them together: f'(g(x)) · g'(x). This calculator does all of this automatically with detailed steps.

Can the chain rule be applied to trigonometric functions?

Yes, the chain rule is frequently used with trigonometric functions. For example: d/dx[sin(3x)] = cos(3x) · 3 = 3cos(3x), d/dx[cos(x²)] = -sin(x²) · 2x = -2x·sin(x²), d/dx[tan(ln(x))] = sec²(ln(x)) · (1/x). Any trig function with an argument that isn't simply x requires the chain rule.

What is the double or nested chain rule?

The nested chain rule (sometimes called the "double chain rule") applies when you have three or more functions composed together, like sin(cos(x²)). You apply the chain rule from the outside in: the derivative of sin(cos(x²)) is cos(cos(x²)) · (-sin(x²)) · 2x. Each layer multiplies by the derivative of the next inner function.

What is the chain rule for exponential functions?

d/dx[e^u] = e^u · u', where u is a function of x. For example: d/dx[e^(x²)] = e^(x²) · 2x, d/dx[e^(sin(x))] = e^(sin(x)) · cos(x). For general exponentials like a^u: d/dx[a^u] = a^u · ln(a) · u'.

How does the chain rule work with the power rule?

When you have a non-trivial expression raised to a power, combine the power rule with the chain rule: d/dx[u^n] = n · u^(n-1) · u'. For example, d/dx[(x²+1)^5] = 5(x²+1)^4 · 2x = 10x(x²+1)^4. This is sometimes called the "generalized power rule."

What is the chain rule for logarithmic functions?

d/dx[ln(u)] = (1/u) · u', where u is a function of x. For example: d/dx[ln(x²+1)] = (2x)/(x²+1), d/dx[ln(sin(x))] = cos(x)/sin(x) = cot(x). For other bases: d/dx[log_a(u)] = u'/(u · ln(a)).

Chain Rule Calculator

f(g(x)) =

= sin(x^2 + 1)

Examples

sin(x²)

cos(3x+1)

e^(x²)

ln(x²+1)

(2x+1)⁵

√(x²+4)

tan(sin(x))

sin³(x)

e^(sin(x))

ln(cos(x))

(x³-2x)⁴

arctan(x²)

Variable

Evaluate at a point

Derivative

Chain Rule

cos(x^2 + 1) · 2x

Chain Decomposition

Outer and inner functions

Outer function

f(u) = sin(u)

f′(u) = cos(u)

Inner function

g(x) = x^2 + 1

g′(x) = 2x

Graph

f(x) and f'(x)

f(x)

f′(x)

Step-by-Step Solution

Chain rule applied step by step

Identify the composite function

f(g(x)) where f(u) = sin(u)

u = g(x) = x^2 + 1

sin(x^2 + 1)

Differentiate the outer function

f′(u) = cos(u)

Substitute u = x^2 + 1

f′(g(x)) = cos((x^2 + 1))

Differentiate the inner function

g(x) = x^2 + 1

Apply standard differentiation rules

g′(x) = 2x

Apply the Chain Rule: multiply derivatives

d/dx[f(g(x))] = f′(g(x)) · g′(x)

= cos((x^2 + 1)) · 2x

cos(x^2 + 1) · 2x

Common Chain Rule Patterns

Quick reference for frequently used compositions

Function	Derivative
sin(u)	cos(u) · u'
cos(u)	-sin(u) · u'
e^u	e^u · u'
ln(u)	(1/u) · u'
u^n	n·u^(n-1) · u'
tan(u)	sec²(u) · u'
√u	1/(2√u) · u'
a^u	a^u·ln(a) · u'
arcsin(u)	1/√(1-u²) · u'
arctan(u)	1/(1+u²) · u'

What Is the Chain Rule?

The fundamental rule for differentiating composite functions

Composition of functions

The chain rule computes the derivative of a composite function — a function built by plugging one into another. If y = f(g(x)), the chain rule gives dy/dx = f'(g(x)) · g'(x).

Leibniz's intuition

If y = f(u) and u = g(x), then dy/dx = (dy/du) · (du/dx). The du terms "cancel" like fractions — an elegant mnemonic that makes the rule easy to remember.

Chain Rule Formula

d/dx[f(g(x))] = f'(g(x)) · g'(x)

Lagrange

(f ∘ g)'(x) = f'(g(x)) · g'(x)

Leibniz

dy/dx = (dy/du) · (du/dx)

Common Chain Rule Patterns

Most frequently used chain rule derivatives in calculus

Pattern	Formula	Example
Trig + Chain	d/dx[sin(u)] = cos(u) · u'	sin(3x) → 3cos(3x)
Exponential + Chain	d/dx[e^u] = e^u · u'	e^(x²) → 2x·e^(x²)
Power + Chain	d/dx[u^n] = n·u^(n-1) · u'	(2x+1)⁵ → 10(2x+1)⁴
Log + Chain	d/dx[ln(u)] = (1/u) · u'	ln(x²+1) → 2x/(x²+1)
Root + Chain	d/dx[√u] = u'/(2√u)	√(x²+4) → x/√(x²+4)
Inverse Trig + Chain	d/dx[arctan(u)] = u'/(1+u²)	arctan(x²) → 2x/(1+x⁴)

Always multiply by the inner derivative u' — it's the step most students forget.

When to Use the Chain Rule

Recognizing composite functions in calculus problems

Use the chain rule whenever you need to differentiate a function within a function. If the argument is more than just the variable, the chain rule applies.

Trig of a non-x expression

sin(x²), cos(3x + 1), tan(ln(x))

Exponential with a non-x exponent

e^(2x), e^(sin(x)), 2^(x²)

Power of a non-x base

(x² + 1)³, (sin(x))⁴

Log of a non-x argument

ln(x² + 1), log(sin(x))

Nested functions

sin(cos(x)), e^(ln(x²)), √(tan(x))

If the argument is simply x (like sin(x) or e^x), use basic derivative rules directly — no chain rule needed.

Common Mistakes with the Chain Rule

Pitfalls to avoid when differentiating composite functions

Forgetting the inner derivative

The most common error — differentiating the outer function but forgetting to multiply by the derivative of the inner function.

Wrong

d/dx[sin(x²)] = cos(x²)

Correct

d/dx[sin(x²)] = cos(x²) · 2x

Not recognizing the composition

Failing to identify that a function is composite and trying to use basic rules directly.

Wrong

d/dx[e^(3x)] = e^(3x)

Correct

d/dx[e^(3x)] = 3·e^(3x)

Wrong order of composition

Mixing up which function is the outer and which is the inner when decomposing.

Wrong

sin(x²): outer = x², inner = sin

Correct

sin(x²): outer = sin(u), inner = x²

Frequently Asked Questions

Common questions about the chain rule and composite function derivatives

The chain rule applies to composite functions (function inside a function), like sin(x²). The product rule applies to products of two functions multiplied together, like x² · sin(x). For sin(x²), use the chain rule. For x² · sin(x), use the product rule. Some problems require both — for example, x² · sin(x²) needs the product rule for the overall structure and the chain rule for sin(x²).

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Last updated Apr 29, 2026

Chain Rule Calculator

Chain Decomposition

Graph

Step-by-Step Solution

Common Chain Rule Patterns

What Is the Chain Rule?

Common Chain Rule Patterns

When to Use the Chain Rule

Common Mistakes with the Chain Rule

Frequently Asked Questions

Embed Chain Rule Calculator

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