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 differentiating polynomials.
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|