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To differentiate composite functions of the form f(g(x)) we use the chain rule (or "function of a function" rule). The derivative of the function of a function f(g(x)) can be expressed as: f'(g(x)).g'(x)
Alternatively if y=f(u) and u = g(x) then \displaystyle \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}

## Summary/Background

The chain rule is also known as the "function of a function" rule. It can be stated in a number of ways.
If y = f(u), u = g(x) then \displaystyle\frac{dy}{dx} = \frac{dy}{du}\times \frac{du}{dx}
Here are some more examples:
\displaystyle \frac{d((x+1)^6)}{dx} = 6(x+1)^5.1 = 6(x+1)^5
\displaystyle \frac{d(e^{10x})}{dx} = e^{10x}.10 = 10e^{10x}
\displaystyle \frac{d( \ln x^2)}{dx} = \frac{1}{x^2}.2x = \frac{2}{x}

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## Glossary

### chain rule

A rule for differentiating a function of a function:
dy/dx = dy/du x du/dx.

### composite

made of a combination of simpler shapes or bodies

### derivative

rate of change, dy/dx, f'(x), , Dx.

### differentiate

to find the derivative of a function

### function

A rule that connects one value in one set with one and only one value in another set.

### rule

A method for connecting one value with another.

### union

The union of two sets A and B is the set containing all the elements of A and B.

Full Glossary List

## This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AP Calculus AB (USA)3DifferentiationRules - the chain rule-
AP Calculus BC (USA)3DifferentiationRules - the chain rule-
AQA A-Level (UK - Pre-2017)C3DifferentiationRules - the chain rule-
AQA A2 Maths 2017Pure MathsDifferentiationChain Rule-
AQA AS/A2 Maths 2017Pure MathsDifferentiationChain Rule-
CBSE XII (India)CalculusContinuity and DifferentiabilityDerivative of composite functions, chain rule-
CBSE XI (India)CalculusLimits and DerivativesSum, difference, product and quotient rules-
CCEA A-Level (NI)C3DifferentiationRules - the chain rule-
CIE A-Level (UK)P2DifferentiationRules - the chain rule-
Edexcel A-Level (UK - Pre-2017)C3DifferentiationRules - the chain rule-
Edexcel A2 Maths 2017Pure MathsDifferentiationChain Rule-
Edexcel AS/A2 Maths 2017Pure MathsDifferentiationChain Rule-
I.B. Higher Level6DifferentiationRules - the chain rule-
I.B. Standard Level6DifferentiationRules - the chain rule-
Methods (UK)M8DifferentiationRules - the chain rule-
OCR A-Level (UK - Pre-2017)C3DifferentiationRules - the chain rule-
OCR A2 Maths 2017Pure MathsDifferentiation TechniquesChain Rule-
OCR MEI A2 Maths 2017Pure MathsDifferentiation TechniquesChain Rule-
OCR-MEI A-Level (UK - Pre-2017)C3DifferentiationRules - the chain rule-
Pre-U A-Level (UK)4DifferentiationRules - the chain rule-
Scottish Advanced HighersM1DifferentiationRules - the chain rule-
Scottish (Highers + Advanced)HM3DifferentiationRules - the chain rule-
Scottish HighersM3DifferentiationRules - the chain rule-
Universal (all site questions)DDifferentiationRules - the chain rule-
WJEC A-Level (Wales)C3DifferentiationRules - the chain rule-