Finding the graph of a derivative involves visually interpreting the slope of the original function's graph at various points. Essentially, you're plotting the rate of change of the original function.
Understanding the Relationship
The derivative of a function, f(x), represents the instantaneous rate of change of f(x) at any given point. Graphically, this rate of change is the slope of the tangent line to the graph of f(x) at that point. Therefore, the graph of the derivative, f'(x), plots these slopes as y-values corresponding to the original x-values.
Steps to Graph a Derivative:
-
Identify Key Points on the Original Graph: Look for points where the original function has:
- Local Maxima/Minima: At these points, the tangent line is horizontal, and the derivative is zero.
- Inflection Points: These are points where the concavity of the original graph changes (from concave up to concave down, or vice versa). They correspond to maxima or minima on the derivative graph.
- Points of Discontinuity: These are points where the original function is not continuous or not differentiable. The derivative might have vertical asymptotes or be undefined at these points.
- Regions of Constant Slope: Regions where the original function is linear. The derivative will be a horizontal line in these regions.
-
Determine the Slope at Key Points:
- Estimate or calculate the slope of the tangent line at each of the key points identified in step 1. Remember that a horizontal tangent line has a slope of zero.
-
Assess Intervals Between Key Points:
- Increasing Function: If the original function is increasing over an interval, the derivative will be positive in that interval. The steeper the increase, the larger the positive value of the derivative.
- Decreasing Function: If the original function is decreasing over an interval, the derivative will be negative in that interval. The steeper the decrease, the larger the negative value of the derivative (i.e., the more negative).
- Concavity:
- Concave Up: The derivative is increasing.
- Concave Down: The derivative is decreasing.
-
Plot the Derivative:
- Use the information gathered to plot points on the derivative graph. The x-coordinates will be the same as on the original graph, and the y-coordinates will be the estimated slopes.
- Connect the points to create a smooth curve (unless there are discontinuities).
Example
Let's say you have a graph of a parabola opening upwards (f(x) = x²).
- Key Points: The vertex of the parabola (at x=0) is a minimum point.
- Slope at Key Points: The slope at the vertex (x=0) is 0.
- Intervals:
- For x < 0, the parabola is decreasing, so the derivative will be negative. The slope becomes increasingly negative as you move further left.
- For x > 0, the parabola is increasing, so the derivative will be positive. The slope becomes increasingly positive as you move further right.
- Plotting: The derivative will be a straight line passing through the origin with a positive slope (f'(x) = 2x).
Summary Table
| Feature of Original Function f(x) | Corresponding Feature of Derivative f'(x) |
|---|---|
| Local Maximum | f'(x) = 0 and f'(x) changes from positive to negative |
| Local Minimum | f'(x) = 0 and f'(x) changes from negative to positive |
| Increasing | f'(x) > 0 |
| Decreasing | f'(x) < 0 |
| Constant | f'(x) = 0 |
| Concave Up | f'(x) is increasing |
| Concave Down | f'(x) is decreasing |
By understanding the relationship between a function and its derivative, and by carefully analyzing the original function's graph, you can sketch a reasonably accurate graph of its derivative.
