quotient rule: given a valid quotient expression(Call it top over bottom[top/bottom]
[(bottom)(top')  (top)(bottom')]/bottom^2
our given expression is such that
top = 8^{1/4}
bottom = x^{8}
Substituting yields the following:
[(bottom)(top')  (top)(bottom')]/bottom^2 ===> [(x^{8})(d(8^{1/4}))  (8^{1/4})(d(x^{8})]/(x^{8})^{2}
(Are you with me?)
what is the derivative of a constant? 0! Now the expression becomes
[0  (8^{1/4})(d(x^{8})]/(x^{8})^{2}
First, what is the value of the denominator? (Multiply the exponents in that expression yields, x^{16}.
This yields, [ (8^{1/4})(d(x^{8})]/x^{16}
Finally, what is the derivative of x^{8}
The formula is as follows :
f(x) = X^{N}; then f^{'}(x) = NX^{N1}; applying yields ==> [ (8^{1/4})(8x^{7})]/x^{16}
doing the rest of the calculations yields(This is calculus, and I KNOW you can do this; just convince YOURSELF you can. You are taking
CALCULUS; You are among the ELITE)
Adding the exponents of 8 yields : 8^{5/4}
subtracting the exponents of X yields : X^{9}
ANS:  8^{5/4}/X^{9}
10/4/2013

Harry D.