RADICALS
To
differentiate a function containing a radical, replace the radical by a
fractional exponent; then find the derivative by applying the appropriate
theorems.
EXAMPLE.
Find the derivative of
SOLUTION:
Replace the radical by the proper fractional exponent, such that
and
by Theorem 6
EXAMPLE:
Find the derivative of
SOL UTION: Replace the radical by the proper fractional exponent,
thus
At
this point a decision is in order. This problem may be solved by either writing
and applying Theorem 6 in the denominator and then
applying Theorem 5 for the quotient or writing
and applying Theorem 6 for the second factor and then
applying Theorem 4 for the product.
The two methods of solution are completed individually as
follows:
Use equation (1):
Find the derivative of the denominator
by applying the power theorem
^{}
The derivative of the numerator is
Now apply Theorem 5:
Multiply both numerator and denominator by
and simplify:
To find the same solution by a different method, use equation
(2):
Find the derivative of each factor:
and
^{}
Now apply Theorem 4:
Multiply both numerator and denominator by
such that,
which agrees with the solution of the first method used.
PRACTICE
PROBLEMS:
Find
the derivatives of the following:
ANSWERS:
