This is a page that will guide you though how to graph derivatives of other graphs. If you have any ideas of how to improve this page, please e-mail them to Marci@RegalLessons.com.
Original Equation (y)
Critical Points: These occur where the slope is 0. They are the minimums and maximums of your graph.
Points of Inflection: These occur where concavity changes in the original.
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First Derivative (y’)
The first derivative is the graph of the slopes of the original equation.
How to Graph
Step 1: Critical points (maximums and minimums) of the original equation are where the zeros are now the zeros (y’ = 0). Plot those points.
Step 2: Where the slope is positive in the original, y’ is positive. Draw the positive parts of the y’ graph with the maximums being where points of inflection were in y.
Step 3: Where the slope is negative in the original, y’ is negative. Draw the negative parts of the y’ graph with the minimums being where points of inflection were in y.
Where are they now?
Original Critical Points: The x-values from the original critical points are the first derivative zeros (y’ = 0).
Original Points of Inflection: The x-values of the original points of inflection are the y’ critical points (maximums and minimums).
Original Decreasing Slope: Where the original equation has a decreasing or negative slope, the y’ has y-values less than 0 ( y’ < 0).
Original Increasing Slope: Where the original equation has and increasing or positive slope, the y’ has y-values that are positive (y’ > 0).
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Second Derivative (y”)
The second derivative is a graph of the slope of the first derivative.
How to Graph
Follow the same steps as for graphing the first derivative, except use the first derivative graph like it was the original. The second deriviatve is just the derivative of the first derivative.
Step 1: The critical points (maximums and minimums) of y’ are where y” = 0. Plot those points.
Step 2: Where the slope is positive in y’, y” is positive. Draw the positive parts of the y” graph with the maximums being where points of inflection were in y’.
Step 3: Where the slope is negative in the y’, y” is negative. Draw the negative parts of the y” graph with the minimums being where points of inflection were in y’.
Where are they now?
Original Critical Points/ Y’ Zeros: The x-values from the original critical points and the y’ zeros are not significant in the graph of the second derivative.
Original Points of Inflection/ Y’ Critical Points: The x-values of the original points of inflection and the y’ critical points are the x-values for the zeros in the second derivative (y” = 0).
Y’ Point of Inflection: The x-value of the first derivative’s point of inflection is now a critical point (minimum or maximum) in the second derivative.
Original Concave Up: Where the original equation was concave up, the y” is concave down.
Original Concave Down: Where the original equation was concave down, the y” is concave up.
Summary
Points of inflection become critical points. Critical points become zeros.
Asymptotes stay in the same spots.
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