How to plot a graph of the derivative of a function.
Goal:
Theory:
Part 1
The ‘derivative’ or ‘derived function’ evaluates to the gradient of the original function at any given ‘x’ value.
Derivative graphs can tell you a lot about a function. The following graph shows the original function (blue) and its derivative (red) plotted together:
Consider the function plotted in blue above:
y = 0.5×2 – x – 1.5
Using the rules of calculus, the derived function is:
dy/dx = 2×0.5x2-1 – 1×x1-1 – 0
= x – 1
For any ‘x’ value, the ‘y’ value of the derived function equals the gradient of a tangent to the original function at that point. Here are some brief observations from the graph above:
| Original function: y = 0.5x² – x – 1.5 | Derived function: dy/dx = x – 1 |
| For ‘x’ values less than x=1, the original function (blue) is decreasing left to right, so all tangent lines will have a negative gradient. | For ‘x’ values less than x=1, the derived function (red) is negative. |
| At x = 1, the original function has a turning point where the gradient is zero. | At x = 1, the derived function is zero. |
| For ‘x’ values greater than zero, the original function is increasing left to right, so all tangent lines will have a positive gradient. | For ‘x’ values greater than x = 1, the derived function is positive. |
Also, the derived function ‘x – 1’ is a straight line, which indicates that the gradient of the original function increases at a constant rate as ‘x’ increases.
For any function that can be graphed in Maths Helper Plus, a derivative graph can be calculated and plotted automatically. This works even if you don’t know how to calculate the derivative using algebra.
Download the free support file… We have created a Maths Helper Plus document containing the completed example from this topic. You can use this to practice the steps described below, and as a starting point for solving your own problems.
File name: ‘Derivative graph.mhp’ File size: 3kb
Click here to download the file.If you choose ‘Open this file from its current location’, then Maths Helper Plus should open the document immediately. If not, try the other option: ‘Save this file to disk’, then run Maths Helper Plus and choose the ‘Open’ command from the ‘File’ menu. Locate the saved file and open it. If you do not yet have Maths Helper Plus installed on your computer, click here for instructions.
Method:
Part 2
Maths Helper Plus can graph the derivative of a function, as well as create a table of derivative values.
Step 1 Start with an empty Maths Helper Plus document
If you have just launched the software then you already have an empty document, otherwise hold down ‘Ctrl’ while you briefly press the ‘N’ key.
Step 2 Graph your function
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- Press the F3 key
- Type the function
- Press ENTER
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If the graph of the function does not appear, you need to adjust the graph scale.
First reduce the graph scale. Press the F10 key enough times until the main parts of the graph are visible.
To enlarge the graph, hold down ‘Ctrl’ while you press F10.
For more help on entering functions, see the ‘Easy Start’ tutorial in the Maths Helper Plus ‘help’. To access the tutorial, click ‘Help’ on the Maths Helper Plus menu bar, then select: ‘Tutorial…’
Step 3 Display the options dialog box for the function
Carefully point to the plotted function graph line with the mouse and double click. This will display the function options dialog box, like this:
If a different dialog box displays, click its cancel button to close it, then try again.
As shown in the picture above, click with the mouse to select the ‘dy/dx’ option. This causes the derivative graph to be displayed.
If you want to display a graph of the second derivative, then select the ‘dy/dx’ option as well.
OPTIONAL:
To create a table of function and derivative values, select the ‘Data Table’ tab at the top of this dialog box. Once this tab is selected, click the ‘Help’ button at the bottom of the dialog box for further instructions. Note that you won’t need to type anything in the bottom edit box. Leave it as ‘y’.
Click the OK button to close.
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