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 | - |