The following problems extend and augment the material presented in the text. More...

The following problems extend and augment the material presented in the text.

More About Differentiation Formulas

Derive the Quotient Rule from the Product Rule as follows.

a. Define the quotient to be a single function,

b. Multiply both sides by g(x) to obtain the equation Q(x) · g(x) = f (x).

c. Differentiate each side, using the Product Rule on the left side.

d. Solve the resulting formula for the derivative Q’(x).

e. Replace and show that the resulting formula for Q’(x) is the same as the Quotient Rule.

Note that in this derivation when we differentiated Q(x) we assumed that the derivative of the quotient exists, whereas in the derivation on pages 135–136 we proved that the derivative exists.