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.