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:
