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

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> 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      Log On


   

Question 1005760: Did I do this logarithmic differentiation correctly?
f(x) = +%28+%28%28x%2B6%29%5E8%284x-1%29%29%2F%28%288x%2B6%29%5E3%29+%29%5E%281%2F4%29+
+%281%2F4%29+%28ln%28%28x%2B6%29%5E8%29+%2B+ln%284x-1%29+-+ln%28%288x%2B6%29%5E3%29+%29+

how would this be simplified down from this point?
Thanks
Answer by jim_thompson5910(35256) About Me  (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
g%28x%29+=+%28x%2B6%29%5E8
h%28x%29+=+4x-1
k%28x%29+=+%288x%2B6%29%5E3
m%28x%29+=+g%28x%29%2Ah%28x%29+=+%28x%2B6%29%5E8%2A%284x-1%29
n%28x%29+=+%28m%28x%29%29%2F%28k%28x%29%29

Based on those definitions, we know that

















I used the product rule to compute the derivative of m(x).

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Then we compute the derivatie of n(x) using the quotient rule















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Now onto the function f(x). Let's compute the derivative.










It's definitely a messy derivative, but it is the correct answer. I confirmed it with a graphing calculator and computer algebra system (CAS)