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}

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}

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}

## Software/Applets used on this page

## Glossary

### chain rule

A rule for differentiating a function of a function:

dy/dx = dy/du x du/dx.

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.

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic |
---|---|---|---|

AP Calculus AB (USA) | 3 | Differentiation | Rules - the chain rule |

AP Calculus BC (USA) | 3 | Differentiation | Rules - the chain rule |

AQA A-Level (UK - Pre-2017) | C3 | Differentiation | Rules - the chain rule |

AQA A2 Maths 2017 | Pure Maths | Differentiation | Chain Rule |

AQA AS/A2 Maths 2017 | Pure Maths | Differentiation | Chain Rule |

CBSE XII (India) | Calculus | Continuity and Differentiability | Derivative of composite functions, chain rule |

CBSE XI (India) | Calculus | Limits and Derivatives | Sum, difference, product and quotient rules |

CCEA A-Level (NI) | C3 | Differentiation | Rules - the chain rule |

CIE A-Level (UK) | P2 | Differentiation | Rules - the chain rule |

G Differentiation | G4 Differentiation Techniques | Chain Rule | |

Edexcel A-Level (UK - Pre-2017) | C3 | Differentiation | Rules - the chain rule |

Edexcel A2 Maths 2017 | Pure Maths | Differentiation | Chain Rule |

Edexcel AS/A2 Maths 2017 | Pure Maths | Differentiation | Chain Rule |

I.B. Higher Level | 6 | Differentiation | Rules - the chain rule |

I.B. Standard Level | 6 | Differentiation | Rules - the chain rule |

Methods (UK) | M8 | Differentiation | Rules - the chain rule |

OCR A-Level (UK - Pre-2017) | C3 | Differentiation | Rules - the chain rule |

OCR A2 Maths 2017 | Pure Maths | Differentiation Techniques | Chain Rule |

OCR MEI A2 Maths 2017 | Pure Maths | Differentiation Techniques | Chain Rule |

OCR-MEI A-Level (UK - Pre-2017) | C3 | Differentiation | Rules - the chain rule |

Pre-U A-Level (UK) | 4 | Differentiation | Rules - the chain rule |

Scottish Advanced Highers | M1 | Differentiation | Rules - the chain rule |

Scottish (Highers + Advanced) | HM3 | Differentiation | Rules - the chain rule |

Scottish Highers | M3 | Differentiation | Rules - the chain rule |

Universal (all site questions) | D | Differentiation | Rules - the chain rule |

WJEC A-Level (Wales) | C3 | Differentiation | Rules - the chain rule |