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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.

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## 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.

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.

### 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.

Full Glossary List

## This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AP Calculus AB (USA)3DifferentiationCurves-
AP Calculus BC (USA)3DifferentiationCurves-
AQA A-Level (UK - Pre-2017)C1DifferentiationCurves-
AQA AS Maths 2017Pure MathsDifferentiationTangents and Normals-
AQA AS/A2 Maths 2017Pure MathsDifferentiationTangents and Normals-
CBSE XII (India)CalculusApplications of DerivativesTangents and normals-
CCEA A-Level (NI)C1DifferentiationCurves-
CIE A-Level (UK)P1DifferentiationCurves-
Edexcel A-Level (UK - Pre-2017)C1DifferentiationCurves-
Edexcel AS Maths 2017Pure MathsDifferentiationTangents and Normals-
Edexcel AS/A2 Maths 2017Pure MathsDifferentiationTangents and Normals-
I.B. Higher Level6DifferentiationCurves-
I.B. Standard Level6DifferentiationCurves-
Methods (UK)M8DifferentiationCurves-
I.B. (MSSL)7DifferentiationCurves-
OCR A-Level (UK - Pre-2017)C1DifferentiationCurves-
OCR AS Maths 2017Pure MathsApplying DifferentiationTangents and Normals-
OCR MEI AS Maths 2017Pure MathsApplying DifferentiationTangents and Normals-
OCR-MEI A-Level (UK - Pre-2017)C2DifferentiationCurves-
Pre-U A-Level (UK)4DifferentiationCurves-