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Graphs of f ( x ) and f ' ( x )

From a graph of a function f(x) we can make a sketched graph of its derivative f ' (x). To do this, we use some things we talked about earlier.

  • If f is decreasing, its slope (and hence its derivative) is negative. If f is increasing, its slope (and hence its derivative) is positive.
  • From drawing tangent lines to f, we can compare relative values of the derivative and tell where the derivative is greatest.
  • If f is a line, its slope is constant.

Be Careful: The word "it" is dangerous.

Look at these two sentences: 

  • It's increasing, so it's positive. 
  • f is increasing, so ' is positive.

The first sentence is unclear. What does "it" mean? There's no way to know. Whenever we use the words "increasing, decreasing, positive, negative," that we are clear about what (f, f ', or something else?) is increasing, decreasing, positive, or negative.

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