A derivative graph shows how the original function is changing at each x-value. When you read the graph of f'(x), you are not looking at the height of f(x), but at the slope of f(x). This matters because slope information tells you where the original function rises, falls, turns around, and changes curvature.
A single graph of f'(x) can reveal the main shape of f(x) without ever seeing f(x) directly.
The key idea is to compare f'(x) with the x-axis. Where f'(x) is positive, f(x) is increasing, and where f'(x) is negative, f(x) is decreasing. When f'(x) crosses zero, f(x) may have a local maximum or local minimum, depending on how the sign changes.
Inflection points of f(x) occur where f'(x) changes from increasing to decreasing or from decreasing to increasing, which means at local extrema of f'(x).
Understanding Calculus: Reading the Graph of the Derivative
The vertical size of a derivative graph tells how steep the original graph is. A value of f prime close to zero means the original function is nearly flat, even if it is still moving upward or downward. A large positive value means a steep upward tilt.
A large negative value means a steep downward tilt. This helps distinguish a gentle climb from a rapid climb. The units matter too.
If f measures distance in metres and x measures time in seconds, then f prime measures metres per second. A derivative is a rate with units built from the original quantities.
A derivative graph can help reconstruct changes in the original function. Start from one known value of f, then track the accumulated effect of the rates shown by f prime. Over a short interval, a positive rate adds to f and a negative rate subtracts from f.
More precisely, the signed area between the f prime graph and the horizontal axis gives the net change in f over an interval. Area above the axis counts positive and area below counts negative. Equal positive and negative areas can produce no overall change, even though the original function moved up and down along the way.
A zero on the derivative graph needs careful interpretation. A graph of f prime can touch the horizontal axis, then stay on the same side. In that case, the original function has a horizontal tangent but does not turn around.
For example, it can flatten briefly while continuing to rise. A derivative may fail to exist at a sharp corner, cusp, or vertical tangent in the original graph.
These points can still be important when finding high or low values. Endpoints of a chosen interval matter as well, because the greatest or least value on that interval can occur there without any special behavior in f prime.
The shape of the derivative graph gives more detail than its position alone. When f prime rises, the slopes of the original graph are becoming more positive. A function might still be falling during this time, but it falls less steeply as its slopes move toward zero.
When f prime falls, the slopes become more negative. The original function might still be rising, though it rises more slowly. This distinction prevents a common mistake.
Increasing slope is not the same thing as increasing function value. It describes how the steepness changes.
Derivative graphs appear in many real situations. A speed graph is the derivative of a position graph. Its signed area gives displacement, while its height gives instantaneous velocity.
In business, the derivative of cost shows the approximate extra cost of making one more item. In science, a temperature derivative shows how quickly heating or cooling occurs. When reading any derivative graph, first label what the horizontal and vertical axes represent.
Then mark zeros, intervals with positive or negative values, unusually large rates, and places where the graph changes direction. Keep rate, total change, and original value separate in your thinking.
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'(c) = 0 and f'(x) changes from positive to negative at c, then f(x) has a local maximum at x = c.
- If f'(c) = 0 and f'(x) changes from negative to positive at c, then f(x) has a local minimum at x = c.
- If f'(x) changes from increasing to decreasing or decreasing to increasing at x = c, then f(x) may have an inflection point at x = c.
- f''(x) is the slope of f'(x), so f''(x) > 0 means f(x) is concave up and f''(x) < 0 means f(x) is concave down.
Vocabulary
- Derivative
- The derivative f'(x) gives the instantaneous rate of change or slope of the original function f(x).
- Increasing interval
- An interval where f(x) rises as x increases, which happens when f'(x) is positive.
- Critical point
- A point in the domain of f where f'(x) = 0 or f'(x) does not exist.
- Local extremum
- A local maximum or local minimum where the function changes from increasing to decreasing or decreasing to increasing.
- Inflection point
- A point where the concavity of f(x) changes from up to down or from down to up.
Common Mistakes to Avoid
- Reading the height of f'(x) as the height of f(x). The value of f'(x) tells the slope of f(x), not the y-value of f(x).
- Assuming every zero of f'(x) is a maximum or minimum. A local extremum occurs only if f'(x) changes sign at that zero.
- Confusing extrema of f'(x) with extrema of f(x). Peaks and valleys on the derivative graph usually indicate possible inflection points of f(x), not turning points of f(x).
- Ignoring intervals and checking only single points. Whether f(x) increases, decreases, or changes concavity depends on the sign or trend of f'(x) over an interval.
Practice Questions
- 1 The graph of f'(x) is positive on (-4, -1), negative on (-1, 3), and positive on (3, 6). On which intervals is f(x) increasing and decreasing?
- 2 Suppose f'(x) crosses the x-axis at x = -2 from negative to positive, and at x = 5 from positive to negative. Classify the local extrema of f(x) at x = -2 and x = 5.
- 3 A graph of f'(x) has a local maximum at x = 1 and a local minimum at x = 4, with no breaks in the graph. Explain what these points suggest about the concavity and possible inflection points of f(x).