Derivative graphs

Derivative Graphs are visual representations of the derivative of a function with respect to its variable. They show how the slope of the tangent line to a curve changes at each point, allowing for an understanding of the rate of change and the behaviour of the original function.

Use this page to revise the following concepts of derivative graphs:

Key Features of Derivative Graphs

A derivative graph is the plot of the derivative \(f^\prime\left(x\right)\) of a function \(f\left(x\right)\). It visually represents the rate of change of \(f\left(x\right)\) at each value of \(x\). The key features  of derivative graphs include:

Slope correspondence:

  • The value of \(f^\prime\left(x\right)\) at a given point corresponds to the slope of the tangent line to the graph of \(f\left(x\right)\) at that point.

Increasing and decreasing intervals:

  • If \(f^\prime\left(x\right) > 0\), the function \(f\left(x\right)\) is increasing (slopes are positive).
    The derivative graph lies above the \(x\)-axis
  • If \(f^\prime\left(x\right) < 0\), the function \(f\left(x\right)\) is decreasing (slopes are negative).
    The derivative graph lies below the \(x\)-axis

Critical points:

  • If \(f^\prime\left(x\right) = 0\), the function \(f\left(x\right)\) has a horizontal tangent line, which could indicate a local maximum, minimum, or a stationary point of inflection
    This point will be an \(x\)-intercept of the derivative graph

Interpreting a Derivative Graph

  • Peaks or troughs on the derivative graph \(f^\prime\left(x\right)\) represent changes in concavity of the original function \(f\left(x\right)\).  Peaks indicate where the graph changes from curving upward to downward, and troughs indicate where the graph changes from curving downward to upward.
  • Zero crossings in \(f^\prime\left(x\right)\) (points where \(f^\prime\left(x\right) = 0\)) correspond to critical points in \(f\left(x\right)\). These are points where the slope of \(f\left(x\right)\) is momentarily zero. Depending on the behaviour of \(f\left(x\right)\) these points could be local maxima, local minima or stationary points of inflection.

The steepness of the graph of \(f^\prime\left(x\right)\) reflects the rate of change of the slope of \(f\left(x\right)\). Steeper sections of \(f^\prime\left(x\right)\) indicate that the slope of \(f\left(x\right)\) is changing rapidly, whereas flatter sections suggest a more gradual change in the slope.

Worked Example

Example 1

Sketch the derivative graph for \(f\left(x\right) = x^3 - 25x + 10\)

A graph of the cubic function f(x) = x^3 - 25x + 10 (blue curve) along with its derivative f'(x) = x^2 - 25 (green curve). The critical points, where f'(x) = 0, are marked with dashed red vertical lines. The derivative graph crosses the x-axis at the points where f(x) has maxima or minima, indicating where the slope of f(x) is zero. Also highlighted is where the derivative changes sign, making a turning point in the derivative graph that corresponds to a point of inflection in f(x).

Example 2

Sketch the derivative graph for \(f(x) = \sin(x); -2 \pi \le x \le 2\pi\)

A graph of a circular function f(x)=sine(x) (blue curve) alongside its derivative f'(x)=cos(x) (green curve). The function has a sinusoidal shape, with peaks and troughs. The derivative is also sinusoidal but phase-shifted, crossing zero at the maxima and minima of f(x). This illustrates how the derivative represents the slope of f(x), being positive on increasing intervals, negative on decreasing intervals, and zero at local maxima and minima.



Use these tabs to view the worked solution for the question above.

You can see the relationship clearly between the graph of the original, in blue, and its derivative in green.
Or, if

\(f(x)= \begin{cases}  x^2+20, &-5\leq x\leq 5 \\ -10x+95, &\phantom{-}5< x \leq10 \end{cases}\)
then
\(f'(x)= \begin{cases}  2x, &-5\leq x\leq 5 \\ -10, &\phantom{-}5< x \leq10 \end{cases}\)

A piecewise function f(x) and its derivative f'(x) are plotted. The function f(x) is defined as f(x) = x^2 + 20 for -5 ≤ x ≤ 5, and f(x) = -10x + 95 for 5 < x ≤ 10. The graph consists of a parabola with a minimum at (0, 20) for the first interval and a linear segment with a negative slope for the second, that begins from x=5 and intercepts the x-axis at x=9.5. The derivative f'(x) is also piecewise, given by f'(x) = 2x for -5 ≤ x ≤ 5, and f'(x) = -10 for 5 < x ≤ 10. The derivative graph shows a line with a positive slope for the first interval and a horizontal line at -10 for the second. There is a discontinuity in f'(x) at x = 5, where the derivative abruptly changes from 10 to -10