Calculus helps us to understand the gradient of a curve.
The derivative gives us a ‘gradient function’ i.e. a formula that will give the gradient at a point on the curve.
The gradient on a curve is different at different points on a curve.
On this graph:
When the gradient is positive.
When the gradient is equal to zero.
When the gradient is negative.
To find the gradient function (derivative) for a model or graph we differentiate the function. The technique of differentiation can be carried out using algebraic manipulation.
In general : For the function then
Khan academy step though this power rule with a number of examples.
The following video will take you through three more examples of differentiation of a polynomial.
Practice this technique on Khan Academy. Hints and further videos are available.
There are three commonly used notation methods for the derivative of a function.
|If then||is the derivative of with respect to|
|If then||is the gradient function of|
|If we differentiate with respect to we get|