In **calculus**, the **first** and **second order derivatives** are important for graphing **functions** and **parametric curves**. Our **graphing calculator** can **calculate** and also graph them.

The **derivative graphing calculator** instantly detects if a function is constant in which case it will return 0. For example, if you type in ** sin(x)^2+ cos(x)^2**, you will get

`0`

since the function is constant throughout its domain (always equal to `1`

).
The form of the **derivative** calculated may look different from but equivalent to what you might expect. For example, the **derivative** of
`f(x) = sin(x)cos(x)`

is calculated as
`f'(x) = cos(x)*cos(x)+sin(x)*-sin(x)`

which is equivalent to
`f'(x) = cos`

^{2}(x) - sin^{2}(x)

Remark: You will notice, as you study in **calculus**, that wherever the graph of the function `f(x)`

is **increasing**, `f′(x)`

is **positive**, and its graph lies above the x-axis on the corresponding interval. Conversely, wherever the graph of `f(x)`

is **decreasing**, `f′(x)`

is **negative**, and its graph lies below the x-axis on the corresponding interval (assuming the axes are not rotated).
Furthermore, you will also observe that wherever the function `f(x)`

is **concave-up**, `f′′(x)`

is **positive**, and its graph lies above the x-axis on the corresponding interval. Conversely, wherever the function `f(x)`

is **concave-down**, `f′′(x)`

is **negative**, and its graph lies below the x-axis on the corresponding interval.