math
posted by bobby on .
Let
f(x)=((x^(6))*((x8)^(2)))/((x2+4)^(7))
Use logarithmic differentiation to determine the derivative.
f '(x) =
f '(8) =

ln f = ln(x^6) + ln(x8)^2  ln(x^2+4)^7
= 6ln(x) + 2ln(x8)  7ln(x^2+4)
1/f f' = d/dx 6ln(x) + 2ln(x8)  7ln(x^2+4)
f ' = (6/x + 2/(x8)  14x/(x^2+4)) * (x^6 * (x8)^2) / (x^2+4)^7
you can massage that in various ways. One way ends up as
2(x8)(x^5)(3x^332x^216x+96)/(x^2+4)^8
plug in x=8 to get the value. That (x8) in the numerator makes it all 0. 
Actually, if you hadn't been asked for f '(x), you could have saved all the work by noting that if (xk)^n is a factor of f(x), and n > 1, then (xk) will be a factor of each term of f '(x), so f '(k) will be zero.