## The Gradient of a Curve

 The gradient of a straight line is a measure of how steep it is. The gradient of a straight line is constant for any point on the line. The gradient of a curve at any point is given by the gradient of the tangent at that point. The gradient of a curve is different at each point on the curve. ### The Gradient Function of y = x²

Consider the curve y = x². Investigate the gradient at various points. (Worked out by finding the slope of the tangents as above.) The gradients themselves form a function:

 x Gradient -2 -4 -1 -2 0 0 1 2 2 4 The equation of this function is y = 2x.

This function is called the gradient function and from it the gradient of the curve y = x² can be found at any point, for any value of x.

The gradient function is sometimes called the derived function or the derivative.

In the next topic, a non-graphical method of finding the derived function will be shown. This process is called differentiation.

Here are three functions (in pink) and their gradient functions (in red).

 Function is y = x² Function is y = 2x² + 3x Function is y = x³   Gradient function is y = 2x Gradient function is y = 4x + 3 Gradient function is 3x² "Can you see the algebraic connection between the function and the gradient function, before the next topic?