Reading f′ from the shape of f
The two curves are not independent pictures. Everything the derivative does is a statement about the shape of the function above it, and four correspondences cover almost every exam question:
- \(f'\) crosses zero where \(f\) has a maximum, a minimum or a flat inflection.
- \(f'\) above the axis means \(f\) is rising; below the axis means falling.
- \(f'\) far from the axis means \(f\) is steep; close to it means \(f\) is nearly flat.
- \(f'\) blows up where \(f\) has a vertical tangent or a pole nearby.
Hover the graph above and a tangent line appears at that point. Its steepness is the height of the green curve directly below — that single visual is the whole relationship.
Sketching f′ when you only have a graph
Being handed a graph of \(f\) with no formula and asked to sketch \(f'\) is a standard exam task. Work through it in this order:
- Mark every turning point of \(f\). Those become the zeros of \(f'\).
- Between them, decide whether \(f\) rises or falls. Rising means \(f'\) sits above the axis, falling below.
- Find the steepest stretch in each interval. That is where \(f'\) reaches its local extreme.
- Check the ends: if \(f\) flattens out, \(f'\) approaches zero.
The result will not be exact, and it does not need to be — a sketch is graded on zeros, signs and rough shape. Type a function above and hide \(f\) with the legend to test yourself against the real answer.
Concavity and the second derivative
The second derivative describes how the first one is changing, which shows up in the curvature of \(f\):
- \(f'' > 0\): the curve holds water — concave up, and \(f'\) is increasing.
- \(f'' < 0\): the curve sheds water — concave down, and \(f'\) is decreasing.
- \(f'' = 0\) with a sign change: an inflection point, where the curvature flips.
Set the order control to 2 to plot \(f\) against \(f''\), or work through the higher-order derivatives section on the calculator page.
Working the graph
- Drag to pan, scroll or pinch to zoom, and click to move the marker.
- Arrow keys pan when the canvas has focus; + and − zoom, 0 resets.
- The legend entries are switches — hide either curve to check your reading of the other.
- Poles are left as gaps rather than joined, so a vertical asymptote never looks like a near-vertical line.
Questions
How do I graph a derivative?
Enter the function above and both curves are drawn together — the function in terracotta, its derivative in green. Nothing is sent anywhere; the sampling happens in your browser.
What does the derivative graph tell me about the original function?
Where \(f'\) crosses zero, \(f\) turns. Where \(f'\) is positive, \(f\) rises. The further \(f'\) is from the axis, the steeper \(f\) is at that point.
How do I sketch f′ from a graph of f?
Mark the turning points of \(f\) as zeros of \(f'\), decide the sign of \(f'\) in each interval between them, then place the extremes of \(f'\) where \(f\) is steepest.
Can I graph the second derivative?
Yes — set the order control to 2 and the plot shows \(f\) against \(f''\), which is what you need for concavity and inflection points.
Why does the curve have gaps?
Both curves are sampled point by point, and where a function is undefined — at a pole, or outside its domain — no point is drawn. Joining across the gap would draw a line that is not part of the graph.