The Chain Rule and the Derivative

1) What is f' = 0?

f'(x) = 0 represents the maximum or minimum of f(x).

2) Identifying Where a Function Increases or Decreases

In order to identify where a function increases or decreases, you must first find the critical values of the function. This will give you the critical points which are possible maximums or minimums. Then you must plug in points before and after the critical points in the derivative. This will tell you if the points are negative or positive. You can use a number line to help visualize this. If the points go from negative to positive, then the function is increasing. If the points go from positive to negative, then the function is decreasing.

3) The Chain Rule

The chain rule is the process of finding the derivative of a composite function. Here are the steps to using the chain rule:
1) Take the derivative of the outside function.
2) Rewrite the interior function.
3) Multiply by the derivative of the inside.

Here is an example where we find the derivative and then the equation of the tangent line:

Find the tangent line of the function f(x) = (2x² + x)³ at the point x = -2

f'(x) = (2x² + x)³
        =  3(2x² + x)² • (4x + 1)
f'(x) = 3(2x² + x)² • (4x + 1)

f(-2) = (2(2)² + -2)³
        = (2(4) + -2)³
          = (8 + -2)³
        = (6)³
          = 216

slope = 3(2(-2)² + -2)² • (4(-2) + 1)
         = 3(6)² • (-7)
         = 3(36) • (-7)
         = 756
Our tangent line is:
y - 216 = 756(x + 2)
y - 216 = 756x + 1512
y = 756x + 1728

4) h(x) = f(g(x))
g(-4) = 5, g'(-4) = 2, and f'(5) = 20. Find h'(-4).

h'(x) = f'(g(x)) • g'(x)

h'(-4) = f'(g(-4)) • g'(-4)

h'(-4) = f'(5) • 2

h'(-4) = 20 • 2

h'(-4) = 40