Blog Post #5

1.  F'(0) Represents a critical point. It is a point on the graph in which there is no slope, where a change in direction occurs. It represents a possible maximum/minimum on the graph.

2.  To find where a function increases and decreases, first find the derivative of the equation. If the derivative (slope) is positive, then the function is increasing. If the slope is negative, the function is decreasing. You can also find a critical point. As you approach the critical point from the left and right, the y value will be increasing or decreasing. This will tell you whether the function is increasing or decreasing as it approaches the critical point.

3. The Chain Rule is the process of finding the derivative of a composite function.
Ex: Find the Derivative of the equation f(x)= (4-2x)^3, and the equation of the tangent line at x=3

Chain Rule: f(x)=g(h(x))                          f '(x)= g '(h(x))h '(x)
f(x)= (4-2x)^3          f '(x)= 3(4-2x)^2*(-2) = -6(4-2x)^2
Find the Tangent Line: f(3)=(4-2(3))^3=-8
(3,-8)

Slope: -6(4-2(3))^2= -24

y+8=-24(x-3)

4.  h(x)=f(g(h)) where g(-4)=5, g'(-4)=2, and f'(5)=20. Find h'(-4).
h'(-4)=f'(5)(2)
h'(-4)=20(2)
h'(-4)=40