Differentiation from First Principles

GraphScribble.jpgThe gradient of a straight line joining the points (x1, y1) and (x2, y2) is found using the formula:


To find the gradient of a curve at a particular point, the gradient of the tangent at that point is found.

The Gradient Function of y = x2

Consider the curve y = x2. Investigate the gradient at various points. (Worked out by using a ruler to judge the line of the tangent at various points and then finding each slope.)


The gradients themselves form a function:



It can be seen that 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.

Click Here to see the gradient function being formed.

The process of finding the gradient function is is called differentiation.

Here is the gradient function (in red) of the function f(x) = x2


Finding the General Gradient Function from First Principles

We will now devise an algebraic method to find the gradient function. This method is said to befrom first principles and involves limits. This is a long method and will be replaced by a rule in the next topic but it is important to be able to follow this method and use it for simple functions.

The graph below shows the function f(x).


The straight line PQ has a gradient of Y12_Differentiation_from_First_Principles_05.gif

We now consider what happens to the gradient of PQ as h gets smaller and smaller. i.e. Q gets nearer to P.

As this happens the gradient of PQ will get closer to the gradient of the tangent at P. Try this with a ruler, keep P fixed and move Q towards P.


If we take this to the limit, as h approaches 0, we will find the gradient of the tangent at P and hence the gradient of the curve at P.

Gradient at P = Y12_Differentiation_from_First_Principles_06.gif

Thus, the gradient function shown as f '(x) is given by:


Finding the Gradient Function for y = x2 from first principles

Using Y12_Differentiation_from_First_Principles_08.gif

Substituting f(x) = x2


Thus the gradient function for f(x) = x2 is f '(x) = 2x