Studying Derivative Graphically

Related calculator: Online Derivative Calculator with Steps

In this note we will talk how to sketch graph of derivative, knowing graph of the function and see graphically when function is not differentiable.

Example. Graph of the function is shown. Sketch graph of the derivative.sketching graph of derivative based on graph of function

We know that derivative at some point is slope of tangent line at that point.

Therefore, by calculating slope of tangent lines at different points, we will be able to sketch graph of derivative.

From A to B slopes are negative, so derivative will be negative on this interval.

From B to C slopes are positive, so derivative is positive here.

Similarly, it can be found that derivative is negative from C to D and positive from D to E.

Note, that at points B, C and D tangent lines are horizontal, thus, their slope is 0.

So, value of derivative at these points is 0.

Now, let's see when function is not differentiable.

We know that when function is differentiable, then it is continuous.

From this we make conclusion that if function is discontinuous then it is not differentiable.

Another two cases is when derivative is infinite (tangent line is vertical) or derivative doesn't exist (one-sided limits are not equal). Last case means that graph of function has "corner", in other words changes direction abruptly as graph of the function $$$y={\left|{x}\right|}$$$ at $$$x=0$$$.

non-differentiable functionsSo, function is not differentiable at $$$x=a$$$ when one of the following holds:

  • function is not continuous at $$${x}={a}$$$.
  • derivative at point $$${a}$$$ is infinite (vertical tangent line).
  • there is a corner at point $$${a}$$$ (one-sided limits exist, but they are not equal).

All three cases are shown on the figure.