The Chain Rule
The chain rule shows how to differentiate the composite functions f(g(x)).
The Chain Rule
To differentiate k (g(x)), differentiate k (x), then replace each x by g(x), and finally multiply by the derivative of g(x). (provided that the derivatives on the right-hand side of the equation exist).
A.1: Find f '(x).
A.2: Find y'.
Generalized Power Rule
By examining the Chain Rule closely, this leads to a generalization of the power rule when differentiating a composite function.
Generalized Power Rule
The Chain Rule as the Generalized Power Rule
A.1: Find g'(x).
A.2: Find y'.
A.3: Find g'(x).
Giant Ball of Oil
A.4: A giant ball of oil was dropped from the 10th floor of Kramerica Industries. Upon impacting the ground, the oil began to expand on the ground in a circular fashion. After t days of impacting the ground, the radius of the oil slick is
feet. How fast is the radius of the oil slick expanding after 2 days?
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