📚 Guide to Calculating and Graphing Derivatives
Our powerful graphing calculator instantly computes the first- and second-order derivatives of a function or parametric expression and can graph them simultaneously.
Calculate Symbolic Derivatives and Graph
You can calculate the symbolic derivatives of any expression in the currently active (in focus) box and graph them.
- Calculate Derivatives: Press the Derivative button. The derivatives are displayed on the screen, along with options to graph them.
- Graph the Derivatives:
- To graph the derivative(s) of a function, press the Graph f(x), f'(x) or Graph f(x), f'(x), f''(x) buttons.
- For a parametric curve, the buttons appear as Graph p(t), p'(t) or Graph p(t), p'(t), p''(t).
The graph(s) of the derivative functions appear in a separate window.
If desired, you can add the calculated derivatives to the multi-input pane by selecting them. They will be appended as highlighted new input panels at the bottom of the pane. Doing so allows you to find the first- or second-order critical points for extrema or inflection points of the original function by pressing Solve.
You can close the derivative window by pressing the Done button.
Notes on Derivatives
The derivative graphing calculator instantly detects if a function is constant in which case it will return 0 as its derivative. For example, entering sin(x)^ 2+ cos(x)^2 yields 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)
Importance in Calculus
In calculus, the first and second order derivatives are important for graphing functions and parametric curves.
When you append the panels containing the computed derivatives, you can use them to find, for example, the first- or second-order critical points of the function by pressing the Solve button.
Observations on Graphs
As you study calculus, you will notice the graph of a function f(x) is:
- increasing wherever f'(x) is positive (i.e., its graph lies above the x-axis).
- decreasing wherever f'(x) is negative (i.e., its graph lies below the x-axis).
- concave-up wherever f''(x) is positive (i.e., its graph lies above the x-axis.
- concave-down wherever f''(x) is negative (i.e., its graph lies below the x-axis).