Posted by Anonymous on Tuesday, July 21, 2009 at 8:41am.
You could construct the tangent at different points, and if the graph is free of kinks and vertical asymptotes, you can join the different points to get a graph of the derivative.
A kink will imply a discontinuity in the derivative, and a vertical asymptote will mean the same thing in the derivative.
I hope this just about covers most situations.
Addendum:
"You could construct the tangent at different points..."
should read:
"You could construct and plot the magnitude of the tangents at different points..."
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