Derivative functions > Derivative
1234Derivative

## Theory

You can approximate the slope of the graph of a function $f$ for a given point value of x using the difference quotient over the interval $\left[x,x+h\right]$ . You let $h$ get closer and closer to $0$ and check if the difference quotient is approaching a certain limit value. If this is the case, then you have found the derivative, the required slope. The difference quotient is defined as:

$\frac{\Delta y}{\Delta x}=\frac{f\left(x+h\right)-f\left(x\right)}{x+h-x}=\frac{f\left(x+h\right)-f\left(x\right)}{h}$

When you divide by $h$ (met $h\ne 0$ ) you are left with an expression that only depends on the value of $x$ as $h$ gets closer to $0$ . (Although the slope in the above graph is positive, $h$ can of course also be negative!)
This is the derivative $\frac{\left(\text{d}y\right)}{\left(\text{dx}\right)}$ for any given x.

The function is accordingly called the $x$ derivative function. You write it as $f\text{'}\left(x\right)$ .

This derivative represents the slope of the graph of the function $f$ for any given $x$ . It is also the slope of the tangent of the graph of $f$ at this value of $x$ .

The graph of $f\text{'}\left(x\right)$ is the graph of the slopes of $f$ .