Example 1: Graph the 1st Derivative, f′(x), Given the Graph of f(x).

Sketch the graph of the 1 ^st Derivative, $f^{\prime }\left(x\right)$, based on the given $f\left(x\right)$ graph.

Step 1: Identify and then boldly label the graph you are working with.

In this example you have been given the original graph, the $f\left(x\right)$ graph.

Step 2: Determine if you are being asked to go move down a level or move up a level.

In this example you are being asked to move down a level. You are starting with the original graph, $f\left(x\right)$, and being asked to move down a level, find the derivative graph, $f^{\prime }\left(x\right)$.

Step 3 (If you are moving down ): Mark your given graph with plus (+), minus (-), and zero (0) based on the 1 ^st Derivative Triangle.

Step 4 (If you are moving down ): Use your plus (+), minus (-), and zero (0) that you marked along with their estimated slope to determine the y-values on the graph you are moving down to.

The first points that you will always want to graph will be the zeros that you have marked. You know where a zero value will always be graphed, it will always be graphed on the x-axis . After that you want to work your way left to right keeping in mind that the steeper the graph the larger the number, and increasing means a positive slope, and decreasing means a negative slope. I find it helpful to use approximated values to keep track of it all. A steep positive slope I would use m=15 , or a flat negative slope I use m=-1 in my head or writing on the actual graph. If you are given a piecewise graph with constants and linear equations , you can actually use real values if needed. The derivative of a constant is always zero, and the derivative of a linear equation ( y= m x+b ) is always the m , the slope.

$\left(x,y\right)=\left(x,\mathit{slope} \mathit{of\; the}\mathit{ Original\; Graph}\right)$

$\left(x,y\right)=\left(x,m\right)$

Final Result: In this example you would draw a final version of your 1 ^st Derivative graph based on the sketch from Step 4 .

Remember to always make sure your final answer passes the vertical line test to ensure it is a function. The sketch of the derivative, $f^{\prime }\left(x\right)$,given this starting graph, $f\left(x\right)$,would look like: