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Derivatives

Derivatives

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

From a graph of a function f(x) we can sketch 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," we need to be clear about what (f, f ', or something else?) is increasing, decreasing, positive, or negative.

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