* Site-Outage Notice: Our engineering elves will be tweaking the Shmoop site from Monday, December 22 10:00 PM PST to Tuesday, December 23 5:00 AM PST. The site will be unavailable during this time.
Dismiss
© 2014 Shmoop University, Inc. All rights reserved.
Derivatives

Derivatives

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.

Advertisement
Noodle's College Search
Advertisement
Advertisement
Advertisement