This is a sample from the Maths Helper Plus on-line help applications...
Differentiate from first principles
This topic shows how to use Maths Helper Plus to differentiate a function y = f(x) numerically from first principles.
The derivative of a function y = f(x) at the point (x,f(x)) equals the gradient of the tangent line to the graph at that point. It can be defined as:
where 'h' approaches zero as a limit. This diagram illustrates this concept graphically:
The derivative formula (above) gives the gradient of the secant line between the two points. As the value of 'h' gets smaller, the two points get closer and the gradient of the secant approaches that of the tangent line to the curve at (x,f(x)):
1. Load the 'Differentiate from first principles.tpl' template file
Note: It is recommended that you begin with an empty Maths Helper Plus document before proceeding further. You can create a new empty document by holding down a Ctrl key and pressing the 'N' key.
2. Enter the function to differentiate
Tip: The graph scale is already set up for the function y = x² - 2x + 1. If you just want to see the demonstration, then leave this function as it is.
3. Calculate the derivatives
Symbols from the gradient formula used in this demonstration:
Data sets are as follows:
Return to step '2' above and enter other functions. Investigate the form of the derivative graph and how it relates to the original function. Examples of functions to investigate would be: y=x³, y=sinx, y=cosx.Â