Graphical Derivatives: Tangent Slope Visualizer
The graph of the derivative f-prime plots the slope of f at every x-value, so f-prime is positive where f is increasing, negative where f is decreasing, and zero where f has a horizontal tangent (a peak, valley, or flat spot).
The most common confusion students have when trying to visualize or understand derivatives on a graph itself is the distinction between the actual function f(x) and the derivative function f'(x) itself. This interactive visualization is designed to help.
For any given x value, you can drag the green number to one of the two equations above:
Drag to the f(x) equation (on the left)
• You'll find the height (or y-value) of the white graph at that x-value.
Drag to the f'(x) equation (on the right)
• You'll find the slope at that x-value. You'll see a tangent line drawn on the graph when you drag it to the correct equation.
How do you read a derivative graph from the graph of f?
Graphical derivative worked example: sketching f-prime from f
If a function is given by f(x) = x³ - 3x, identify the x-intercepts and signs of its derivative f'(x) graphically.
1. Find the horizontal tangents: f'(x) = 3x² - 3 = 0 ⟹ x = ±1. These points map to intercepts on the f'(x) graph.
2. For x < -1, the slope is positive since the function increases.
3. For -1 < x < 1, the slope is negative since the function decreases.
4. For x > 1, the slope is positive. Thus, f'(x) is a parabola that is positive, dips below the x-axis between -1 and 1, and rises above it again.
Another worked example
For f given by a parabola opening upward with its vertex at x = 1, describe f-prime. Step 1: f decreases for x less than 1, so f-prime is negative there. Step 2: f has a horizontal tangent at x = 1, so f-prime equals 0 at x = 1. Step 3: f increases for x greater than 1, so f-prime is positive there, giving an f-prime graph that is a straight line crossing zero at x = 1.
Aligned with the College Board AP Calculus CED — Unit 2 (Differentiation: Definition and Fundamental Properties), Topic 2.2, connecting to the analysis in Unit 5. College Board ↗
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