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PRODUCTS Theorem 4. The derivative of the product of two differentiable
functions of x is equal to the first function multiplied by the derivative of
the second function, plus the second function multiplied by the derivative of
the first function. If
then
This theorem may be extended to include the product of
three differentiable functions or more. The result for three functions would be
as follows: if
then
EXAMPLE. Find the derivative of
SOLUTION. The derivative of the first factor is 2x, and
the derivative of the second factor is 4x3. Therefore,
EXAMPLE. Find the derivative of
SOLUTION. The derivatives of the three factors, in the
order given, are 3x2, 2x, and 4x3. Therefore,
Expanding,
we get
PRACTICE
PROBLEMS: Find
the derivatives of the following:
ANSWERS:
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