5.7 The Graph of The Derivative

1. Sketching $f'(x)$ from $f(x)$

If we know the graph of $f(x)$, we can sketch the graph of $f'(x)$. To do this:

  • Spot the stationary points; they become roots.
  • Increasing sections have positive $f'(x)$ (above the x-axis).
  • Decreasing sections have negative $f'(x)$ (below the x-axis).

Furthermore:

  • Points of inflexion (POI) become maximums or minimums on the derivative graph.

In the same way, we can derive the graph of $f''(x)$ from $f'(x)$.

Notice the behavior of a stationary POI ($f'=0$ and $f''=0$)

f f'

EXAMPLE 1

The following example contains 3 stationary points. The third is a stationary POI.

f f' f''

2. Sketching $f(x)$ from $f'(x)$

Given the graph of $f'(x)$, we may need to sketch the graph of $f(x)$. To do this:

  • Spot the roots; they become stationary points.

In particular:

  • A root crossing from − to + corresponds to a minimum.
  • A root crossing from + to − corresponds to a maximum.

A key challenge is that we do not know the vertical position of the maximum or minimum. Without additional information, we place them at arbitrary y-values.

EXAMPLE 2

f' f max poi min

Any vertical translation of the $f(x)$ graph is a valid solution, so the curve could be lower or higher than mapped above.

Often, we are given extra information (e.g., an initial starting point) to help us position the maximum and minimum before sketching the graph.