Transformations of Functions

Let f (x) be some function and let transform be another function that is a transformation of f. This applet explores how the derivatives of f and g are related.

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See About the calculus applets for operating instructions.

1. Transformation of a cubic

The applet initially shows the graph of a cubic. The function and its derivative are in magenta, while the transformation and its derivative are in blue. You can only see the blue one, since they start out identical.

2. A sinusoid

Select the second example, a sinusoid, and experiment with the sliders. In particular, set a = 1, c = d = 0, and play with the b slider. This example more clearly shows that the derivative of the transform has both a vertical and a horizontal stretch/squeeze.

This could have been done by just taking the derivative of g(x), using the chain rule: deriv of transform Notice that d is gone, so vertical shifts don't affect the derivative. c is in the same place, so horizontal shifts also shift the derivative. A is in the same place, so a vertical stretch/squish also happens to the derivative. Notice that b is now in two places, so figuring out the change in the derivative involves both a vertical and a horizontal stretch/squeeze.

Other differentiation topics