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| Powers of Functions | |||||
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POWERS OF FUNCTIONS Theorem 6. The derivative of any differentiable function
of x raised to the power n, where n is any real number, is equal to n times the
polynomial function of x to the (n - 1) power times the derivative of the
polynomial itself. If
where u is any differentiable function of x, then
EXAMPLE. Find the derivative of the function
SOLUTION.- Apply Theorem 6 and find
EXAMPLE. Find the derivative of the function
SOL UTION.- This problem involves Theorem 5 and Theorem 6.
Theorem 6 is used to find the derivative of the numerator; then Theorem 5 is
used to find the derivative of the resulting quotient. The
derivative of the numerator is
and
the derivative of the denominator is 1. Then, by Theorem 5
PRACTICE
PROBLEMS: Find
the derivatives of the following:
ANSWERS:
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