How To Draw The Derivative Graph From The Original

What if I told you that at that place is a way to accept the graph of the derivative and quickly draw the graph of the original function?

Jenn (B.S., M.Ed.) of Calcworkshop® teaching derivative graphs

Jenn, Founder Calcworkshop^®, 15+ Years Experience (Licensed & Certified Teacher)

Well, the secret to understanding a graph lies in properly labelling it and learning how to read it.

Merely it'south all-time to learn how through exploration.

Derivative Graph Rules

Beneath are 3 pairs of graphs. The top graph is the original office, f(x), and the lesser graph is the derivative, f'(ten).

graph of derivative to original function

Graph Of Derivative To Original Function

What do you notice virtually each pair?

If the slope of f(ten) is negative, then the graph of f'(ten) will exist beneath the x-centrality.
If the slope of f(ten) is positive, then the graph of f'(x) will exist higher up the x-axis.
All relative extrema of f(ten) volition go x-intercepts of f'(x).
All points of intersection of f(x) volition become relative extrema of f'(x).

Additionally, if f(x) is an odd office, then f'(10) is an even office. And if f(ten) is an even function, then f'(x) is an odd role. This means that the derivative volition more than likely have i less plow than the original function.

Cool, right?

So, graphing the derivative when given the original function is all nigh approximating the slope.

How To Read Derivative Graphs

Alright, this seems unproblematic enough, but what do nosotros do if we are given the derivative graph, and nosotros want to find the original part?

Then glad you asked!

Once once again, you just need to know what to look for!

\brainstorm{equation}
\begin{array}{|l|l|50|}
\hline f^{\prime}(x)>0 & \rightarrow & f(ten) \text { is increasing } \\
\hline f^{\prime number}(x)<0 & \rightarrow & f(x) \text { is decreasing } \\
\hline f^{\prime}(x) \text { changes from negative to positive } & \rightarrow & f(x) \text { has a relative minimum } \\
\hline f^{\prime}(x) \text { changes from positive to negative } & \rightarrow & f(10) \text { has a relative maximum } \\
\hline f^{\prime}(ten) \text { is increasing } & \rightarrow & f(x) \text { is concave upward } \\
\hline f^{\prime}(x) \text { is decreasing } & \rightarrow & f(ten) \text { is concave down } \\
\hline f^{\prime number}(ten) \text { has an extreme value } & \rightarrow & f(x) \text { has a point of intersection } \\
\hline
\end{array}
\terminate{equation}

Let'due south brand sense of this tabular array with a picture. Again, the key to understanding how to analyze the graph of the derivative is to marker upward the graph, as indicated below.

how to find original function from derivative graph

How To Find Original Office From Derivative Graph

estimated function graph

Estimated Function Graph

With the help of numerous examples, we will be able to plot the derivative of an original function and analyze the original function using the graph of the derivative.

Trust me, it's straightforward, and y'all'll go the hang of it in no fourth dimension.

Let's go to it!