← Calculus methods

Sketching between a function and its gradient function

A recurring task in some form nearly every year: given the graph of f(x), sketch f'(x) — or the reverse. This is read directly off the SHAPE, without ever finding an equation.

Reading f'(x) from the shape of f(x)

  • Wherever f(x) has a TURNING POINT (a peak or trough), f'(x) crosses the x-axis (is zero) at that same x-value.
  • Wherever f(x) is INCREASING, f'(x) is POSITIVE (above the x-axis); wherever f(x) is DECREASING, f'(x) is NEGATIVE (below the x-axis).
  • The STEEPER f(x) is at a point, the FURTHER f'(x) is from zero at that same x-value — a very steep section of f(x) produces a large positive or negative value on f'(x), not just a sign.
  • A cubic f(x) with two turning points produces a QUADRATIC-shaped f'(x) (one degree lower); a quartic f(x) produces a cubic-shaped f'(x).
Worked example Sketching f'(x) from a described cubic A cubic f(x) has a local maximum at x = −3 and a local minimum at x = 3, and is increasing everywhere else. Describe the shape of f'(x). f'(x) = 0 at x = −3 and x = 3, since both are turning points of f(x). f(x) is increasing for x < −3, so f'(x) is POSITIVE there. f(x) is decreasing between x = −3 and x = 3 (from a local max down to a local min), so f'(x) is NEGATIVE in that interval. f(x) is increasing again for x > 3, so f'(x) is POSITIVE there. A cubic f(x) gives a quadratic-shaped f'(x): this matches a parabola opening upward, crossing zero at x = −3 and x = 3, negative between them and positive outside. ✓ f'(x) is an upward parabola crossing the x-axis at x = −3 and x = 3, negative in between and positive either side — degree one lower than f(x), as it always is. ✕

A turning point on f(x) does not mean a turning point on f'(x) at the same place

A turning point of f(x) becomes a ZERO (an x-intercept) of f'(x), not a turning point of f'(x) itself. Confusing "where f(x) turns" with "where f'(x) turns" is a common sketching error — always ask separately where f(x) is flat (giving f'(x) = 0) and where f(x) is steepest (giving the extreme values, positive or negative, of f'(x)).

Practise just this