# Math Insight

### Developing intuition about the derivative

The derivative of a piecewise linear function. The blue line segments are the graph of a function $f(x)$ that is linear along each of a bunch of small intervals in $x$. You can change $f$ by dragging the blue points, which move the ends of line segments up and down. You can also drag the red points to move a line segment up and down without changing its slope. The derivative $f'(x)$ of the function $f(x)$ is shown by the green horizontal line segments. The derivative $f'(x)$ indicates the slope of the function $f(x)$. Since, along each small interval of $x$, the function $f(x)$ has the same slope, the derivative $f'(x)$ is constant along each of those intervals. If two adjacent line segments of $f(x)$ have two different slopes, then the derivative $f'(x)$ jumps to a new value at the point between the corresponding intervals in $x$. To test your ability to estimate the derivative from the function, you can uncheck the “show derivative” checkbox and attempt to sketch what you think the derivative is. Alternatively, you can uncheck the “show function” checkbox to test your ability to sketch the function from its derivative.

Derivative of interpolating polynomial. The blue curve is the graph of a polynomial $f(x)$. You can change $f$ by dragging the blue points, as $f$ is an interpolating polynomial through those points. The derivative $f'(x)$ of the function $f(x)$ is shown by the green curve. The derivative $f'(x)$ indicates the slope of the function $f(x)$, so that it is positive when $f$ is increasing, negative when $f$ is decreasing, and zero at the points where the tangent line to $f$ is horizontal. To test your ability to estimate the derivative from the function, you can uncheck the “show derivative” checkbox and attempt to sketch what you think the derivative is. Alternatively, you can uncheck the “show function” checkbox to test your ability to sketch the function from its derivative.

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.