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Derivatives
Derivatives
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Tangent Lines and Derivatives

The following phrases all mean the same thing:

  • the slope of f at a
  • the derivative of f at a
  •  f ' (a)
  • the slope of the tangent line to f at a
  • the instantaneous rate of change of f at a

Since f ' (a) (the derivative of f at a) is equal to the slope of the tangent line to f at a, we can determine the sign of f ' (a) by looking at the tangent line to f at a.

Sample Problem

Consider this function:

The tangent line to f at a is sloping upwards. This means the slope of the tangent line to f at a is positive. Since the slope of the tangent line to f at a equals f ' (a), we know f ' (a) is also positive.

We can also tell the sign of f ' (a) by looking at the function f and not thinking about tangent lines.

  • If f is increasing at a then f ' (a) is positive.
      
  • If f is decreasing at a then f ' (a) is negative.
      
  • If f "flattens out" at a then f ' (a) is zero.

We can also use tangent lines to compare values of f' at different points.

Remember these pictures. We'll use them again when we discuss concavity and second derivatives.

Next Page: Tangent Line Approximation
Previous Page: When Tangent Lines Don't Exist

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