# Calculating First & Second Order Derivatives

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) = cos2(x) - sin2(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.