Applications of differentiation
Read curve shape and derivative graphs · Diagrams
A cubic and its derivative
One input, two different heights
Compare f(x) = 0.1x(x − 6)(x + 6) with f′(x) = 0.3x² − 3.6. The upper height is the function value; the lower height is its slope. Both graphs use the same horizontal scale.
Compare the same inputs
| Input x | Original f(x) | Slope f′(x) | Interpretation |
|---|---|---|---|
| −2√3 | 4.8√3 ≈ 8.31 | 0 | Local maximum; derivative crosses down through zero. |
| 0 | 0 | −3.6 | The original crosses the axis while decreasing; it is not stationary. |
| 2√3 | −4.8√3 ≈ −8.31 | 0 | Local minimum; derivative crosses up through zero. |
The table and original diagram remain available without loading the explorer. Displayed decimals are approximate.
Teaching and source notes
The source prints approximate turning coordinates (−3.46, 8.31) and (3.46, −8.31). The exact inputs are ±2√3 and the exact original heights are ∓4.8√3. Presets use the exact inputs rather than the printed rounded decimals.
The Teacher answer connects original turning inputs to derivative x-intercepts. At the left point the derivative crosses from positive to negative; at the right it crosses from negative to positive. A derivative zero alone is not a general turning-point test.
At x = 0 the derivative is −3.6, so the original point is not stationary. A point on the axis need not have a horizontal tangent.