The following applet
can be used to illustrate how the chain rule applies to the derivative of
f(g(x)) at x=a. Simply enter the functions g(x) and f(x) and the value a. The
applet automatically draws the graphs of g(x), f(x) and f(g(x)). The
appropriate line tangent to the graph is drawn in red. For g(x), the tangent
line goes through the point (a,g(a)), for f(x), the tangent line goes through
the point (g(a),f(g(a))) and for f(g(x)), the tangent line goes through the
point (a,f(g(a))). The value of a can be changed by simply typing a new value,
such as "1.2345", "pi/2", "sqrt(5)+cos(3)", etc. You may also change this value
by using the up/down arrow keys or dragging the corresponding point left or
right. To move the center of the graph, simply drag any point to a new
location. To label the x-axis in radians (i.e. multiples of pi), click on the
graph and press "control-r". To switch back, simply press "control-r" again.

## Software/Applets used on this page

This page uses an applet from David Little (Lecturer in the Mathematics Department, Penn State University, USA) and is used with his permission.

## 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.

### derivative

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

### graph

A diagram showing a relationship between two variables.

The diagram shows a vertical y axis and a horizontal x axis.

The diagram shows a vertical y axis and a horizontal x axis.

### rule

A method for connecting one value with another.

### tangent

1. The trigonometrical function defined as opposite/adjacent in a right-angled triangle.

2. A straight line that touches a curve at one point.

2. A straight line that touches a curve at one point.

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic | Exam Year |
---|---|---|---|---|

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 | - |

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 | - |