# Math Insight

### Calculating the derivative of a quadratic function

Calculating the derivative of a quadratic function.

In the below applet, you can change the function to $f(x)=3x^2$ or another quadratic function to explore its derivative. To enter $f(x)=3x^2$, you can type 3*x^2 in the box for $f(x)$.

The derivative of a function. The function $f(x)$ is plotted by the thick blue curve. Its derivative $f'(x)$ is shown by the thin green curve. The large red diamond on the graph of $f$ represents a point $(x_0,f(x_0))$, and you can change $x_0$ by dragging this point with your mouse. A tangent line to $f$ calculated at $x=x_0$ is shown by the red line. Its slope is the derivative $f'(x_0)$ of $f$ evaluated at $x=x_0$. This slope is also displayed by the smaller red diamond on the graph of $f'$, which is at the point $(x_0,f'(x_0))$. As you change $x_0$, this smaller diamond representing the slope traces out the graph of the derivative. You can change $f(x)$ by typing a new value in its box. The value of $f'(x)$ is displayed to the right of the box. You can hide items by unchecking the corresponding check boxes in order to test yourself on how well you can determine the derivative from the function or vice versa. You can use the buttons at the top to zoom in and out as well as pan the view.