To find the gradient of a tangent to a curve y=f(x) at a given point x=a:

firstly differentiate the equation of the curve in order to find \displaystyle\frac{dy}{dx} ,

secondly substitute into this expression the value x=a.

## Summary/Background

If y = ax^n then \displaystyle\frac{dy}{dx} = anx^{n-1} and if y = a then \displaystyle \frac{dy}{dx} = 0, where a is a constant.

These work for all values of n, positive or negative, decimal or fraction.

These work for all values of n, positive or negative, decimal or fraction.

## Software/Applets used on this page

## Glossary

### constant

a value in a formula or equation that cannot change.

### differentiate

to find the derivative of a function

### equation

A statement that two mathematical expressions are equal.

### fraction

A ratio of two numbers or polynomials.

### gradient

The slope of a line; the angle of its inclination to the horizontal.

### 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.

### union

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

### work

Equal to F x s, where F is the force in Newtons and s is the distance travelled and is measured in Joules.

## This question appears in the following syllabi:

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

AP Calculus AB (USA) | 3 | Differentiation | Curves | - |

AP Calculus BC (USA) | 3 | Differentiation | Curves | - |

AQA A-Level (UK - Pre-2017) | C1 | Differentiation | Curves | - |

AQA AS Maths 2017 | Pure Maths | Differentiation | Tangents and Normals | - |

AQA AS/A2 Maths 2017 | Pure Maths | Differentiation | Tangents and Normals | - |

CBSE XII (India) | Calculus | Applications of Derivatives | Tangents and normals | - |

CCEA A-Level (NI) | C1 | Differentiation | Curves | - |

CIE A-Level (UK) | P1 | Differentiation | Curves | - |

Edexcel A-Level (UK - Pre-2017) | C1 | Differentiation | Curves | - |

Edexcel AS Maths 2017 | Pure Maths | Differentiation | Tangents and Normals | - |

Edexcel AS/A2 Maths 2017 | Pure Maths | Differentiation | Tangents and Normals | - |

I.B. Higher Level | 6 | Differentiation | Curves | - |

I.B. Standard Level | 6 | Differentiation | Curves | - |

Methods (UK) | M8 | Differentiation | Curves | - |

I.B. (MSSL) | 7 | Differentiation | Curves | - |

OCR A-Level (UK - Pre-2017) | C1 | Differentiation | Curves | - |

OCR AS Maths 2017 | Pure Maths | Applying Differentiation | Tangents and Normals | - |

OCR MEI AS Maths 2017 | Pure Maths | Applying Differentiation | Tangents and Normals | - |

OCR-MEI A-Level (UK - Pre-2017) | C2 | Differentiation | Curves | - |

Pre-U A-Level (UK) | 4 | Differentiation | Curves | - |

Scottish (Highers + Advanced) | HM1 | Differentiation | Curves | - |

Scottish Highers | M1 | Differentiation | Curves | - |

Universal (all site questions) | D | Differentiation | Curves | - |

WJEC A-Level (Wales) | C1 | Differentiation | Curves | - |