The first derivative f'(x) measures the instantaneous rate of change, or slope, of a function f(x). Its sign tells whether the original function is moving upward or downward as x increases. When f'(x) is positive, f(x) is increasing, and when f'(x) is negative, f(x) is decreasing.
This idea is one of the most important links between a function and its derivative graph.
Key Facts
- If f'(x) > 0 on an interval, then f(x) is increasing on that interval.
- If f'(x) < 0 on an interval, then f(x) is decreasing on that interval.
- If f'(x) = 0 at x = c, then f(x) has a horizontal tangent at x = c.
- Critical numbers occur where f'(x) = 0 or f'(x) is undefined.
- On the graph of f'(x), regions above the x-axis mean f(x) is increasing.
- On the graph of f'(x), regions below the x-axis mean f(x) is decreasing.
Vocabulary
- Derivative
- The derivative f'(x) gives the instantaneous rate of change or slope of f(x) at a point.
- Increasing function
- A function is increasing on an interval if its y-values rise as x moves from left to right.
- Decreasing function
- A function is decreasing on an interval if its y-values fall as x moves from left to right.
- Critical number
- A critical number is an x-value in the domain of f where f'(x) = 0 or f'(x) does not exist.
- Sign chart
- A sign chart organizes where f'(x) is positive, negative, zero, or undefined across intervals.
Common Mistakes to Avoid
- Confusing the graph of f with the graph of f' is wrong because the height of f'(x) gives slope information about f(x), not the y-value of f(x).
- Assuming f'(x) = 0 always means a maximum or minimum is wrong because the derivative can touch zero without changing sign.
- Looking only at isolated points is wrong because increasing and decreasing behavior is determined on intervals, not just at single x-values.
- Forgetting that f'(x) below the x-axis means f(x) decreases is wrong because negative derivative values indicate negative slope for the original function.
Practice Questions
- 1 For f'(x) = 2x - 6, find the intervals where f(x) is increasing and decreasing.
- 2 For f'(x) = (x + 1)(x - 4), make a sign chart and determine where f(x) is increasing and decreasing.
- 3 A graph of f'(x) is above the x-axis on (-3, 1), crosses the x-axis at x = 1, and is below the x-axis on (1, 5). Describe the behavior of f(x) on these intervals and explain what may happen at x = 1.