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.
Understanding Calculus: The Sign of the Derivative
A useful way to study a function is to split its domain into intervals at critical numbers. These are the places where the derivative is zero or does not exist. Within one open interval, the derivative often keeps the same sign.
Pick one test value from that interval and find the derivative there. Its sign describes the behavior across that whole interval, provided there are no further critical numbers inside it.
This creates a sign chart. A sign chart is a compact record of where a graph rises and falls, even when drawing the full graph would be difficult.
The most important detail is what happens on the two sides of a critical number. If the derivative changes from positive to negative, the function rises before the point and falls after it. The point is a local maximum.
If the derivative changes from negative to positive, the function falls before the point and rises after it. The point is a local minimum. A derivative of zero alone does not prove either result.
For example, the graph of x cubed has a flat tangent at zero, but it continues rising on both sides. Its derivative is positive on both sides except at the single flat point.
A derivative can fail to exist for several reasons. A graph may have a sharp corner, a cusp, or a vertical tangent. At a corner, such as the point of an absolute value graph, the slope approaching from the left differs from the slope approaching from the right.
There is no single derivative there. Such a point can still be a local minimum or maximum, so it belongs in the sign chart.
Students should first check that the original function is defined at a critical number. A gap or a vertical asymptote can split the domain, and behavior on one side cannot be joined to behavior on the other.
This method appears whenever a changing quantity needs to be optimized. A business model may use a function for profit over time. A physics problem may describe height, position, or energy as time changes.
The derivative identifies intervals of growth or decline, while a sign change identifies a possible turning point. In a real problem, the allowed domain matters as much as the calculus. Time may begin at zero, a length cannot be negative, and a model may only be valid for a limited range.
When reading graphs, keep the function and its derivative separate. A high function value does not mean a positive derivative.
It only means the graph is high. The derivative tells the direction of motion at that location.
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.