Change > Derivative functions
12345Derivative functions

## Theory

Any function $y=f\left(x\right)$ usually has a slope at any point of its curve that is given by the derivative $f\text{'}\left(x\right)$ of that point.

You can now also make a graph of the values of these slopes (derivatives). Here you see the curve of a function (in red) together with the corresponding graph (in blue), the graph given by $f\text{'}$.
The function $f\text{'}$ is called the slope function or derivative function.

If you compare the two curves you can see that:

• the values of the slope function are positive while the original function is increasing;

• the values of the slope function are negative while the original function is decreasing;

• at values of $x$ where the slope function has a value of $0$, the original function has a horizontal tangent; these are often extrema of the original function. .

It is therefore primarily the sign (positive, negative or $0$) of the derivative function that provides information about the curve of the original function.