2.8 Differentiation Rules

Summary

If there had not been easily applied rules for finding the derivative of most functions used in modeling, the derivative would not be as powerful a tool as it has turned out to be. Using the limit definition, general rules are developed for constant multiples of a function; sums, products, reciprocals, quotients, and compositions of functions. Along with the Power Rule, these rules permit the calculation of the derivative of a remarkably large number of functions.

By the end of your studying, you should know:

Differentiability implies continuity, and an example showing that the converse is false.
Constant multiple rule for derivatives.
The Sum Rule (for finding the derivative of a sum of functions).
The Product Rule.
The Reciprocal Rule.
The Quotient Rule.
The Chain Rule.

On-screen applet instructions: The table displayed shows the values of the difference quotients of fg for the given value of h and at various points x. The value of h may be controlled by the slider so that one can investigate the limit of the difference quotients as h → 0. The hide/show button at the bottom will show an additional column of values for comparison with the derivative fg′ + f′g.

Examples

Differentiate h(x) = (x + 2)⁴.

Robert throws a rock into a lake, which creates a circle of ripples which moves away from the point of impact at a constant speed of 50 centimeters per second. What is the rate of change in the area of the circle after 1 second? After 10 seconds? After t seconds?

Find the instantaneous rate of change of the volume of a cube with respect to the length of its edge, x, when x equals 4 inches.