SOLUTION: Did I do this logarithmic differentiation correctly?
f(x) = {{{ ( ((x+6)^8(4x-1))/((8x+6)^3) )^(1/4) }}}
{{{ (1/4) (ln((x+6)^8) + ln(4x-1) - ln((8x+6)^3) ) }}}
{{{ (1/4)(1
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Question 1005760: Did I do this logarithmic differentiation correctly?
f(x) =
how would this be simplified down from this point?
Thanks
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
I'm not sure why you introduced natural logs.
Here's how I would have done it
Let
Based on those definitions, we know that
 = 8(x+6)^7)
 = 4)
 = 24(8x+6)^2)
 = g \ '(x)*h(x) + g(x)*h \ '(x))
 = 8(x+6)^7*(4x-1) + (x+6)^8*4)
 = 4(x+6)^7\left[2(4x-1) + (x+6)\right])
 = 4(x+6)^7(8x-2 + x+6))
 = 4(x+6)^7(9x+4))
I used the product rule to compute the derivative of m(x).
-------------------------------------------------
Then we compute the derivatie of n(x) using the quotient rule
 = \frac{m(x)}{k(x)})
 = \frac{m \ '(x)*k(x) - m(x)*k \ '(x)}{\left[k(x)\right]^2})
 = \frac{4(x+6)^7(9x+4)*(8x+6)^3 - (x+6)^8*(4x-1)*24(8x+6)^2}{\left[(8x+6)^3\right]^2})
 = \frac{4(x+6)^7(8x+6)^2\left[(9x+4)(8x+6)-6(x+6)(4x-1)\right]}{(8x+6)^6})
 = \frac{4(x+6)^7(8x+6)^2\left[72x^2+86x+24-24x^2-138x+36\right]}{(8x+6)^6})
 = \frac{4(x+6)^7(8x+6)^2(48x^2-52x+60)}{(8x+6)^6})
 = \frac{4(x+6)^7(48x^2-52x+60)}{(8x+6)^4})
-------------------------------------------------
Now onto the function f(x). Let's compute the derivative.
 = \left(\frac{(x+6)^8(4x-1)}{(8x+6)^3}\right)^{\frac{1}{4}})
 = \left(n(x)\right)^{\frac{1}{4}})
 = \frac{1}{4}\left(n(x)\right)^{\frac{1}{4}-1}*\frac{d}{dx}(n(x)))
 = \frac{1}{4}\left(n(x)\right)^{-\frac{3}{4}} * n \ '(x) )
 = \frac{1}{4}\left(\frac{(x+6)^8(4x-1)}{(8x+6)^3}\right)^{-\frac{3}{4}} * \left[\frac{4(x+6)^7(48x^2-52x+60)}{(8x+6)^4}\right] )
It's definitely a messy derivative, but it is the correct answer. I confirmed it with a graphing calculator and computer algebra system (CAS)
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