Question 155846
{{{y= (x^2-8)^(1/3)(x^3+1)^(1/2)/(x^6-7x+5)}}}
I've taken logs so that log y= 1/3log(x^2-8) + 1/2log(x^3+1) - log(x^6 -7x + 5)
but i am unsure how to simplify it further.
Any assistance would be appreciated.
<pre><font size = 4 color = "indigo"><b> 
Be sure to take NATURAL logs, abbreviated "ln", not "log",
because "log" means COMMON log, that is, base 10 logs.

{{{y= (x^2-8)^(1/3)(x^3+1)^(1/2)/(x^6-7x+5)}}} 
 
{{{ln(y)=(1/3)ln(x^2-8)+(1/2)ln(x^3+1)-ln(x^6-7x+5)}}}
 
{{{((dy)/(dx))/y = (1/3)(2x/(x-8))+(1/2)(3x^2/(x^3+1))-(6x^5-7)/(x^6-7x+5)}}}

{{{((dy)/(dx))/y = 2x/(3(x-8))+3x^2/(2(x^3+1))-(6x^5-7)/(x^6-7x+5)}}}

Multiply both side by {{{y}}}

{{{(dy)/(dx) = y*(2x/(3(x-8))+3x^2/(2(x^3+1))-(6x^5-7)/(x^6-7x+5))}}}

Finally you must replace y using the original

{{{y= (x^2-8)^(1/3)(x^3+1)^(1/2)/(x^6-7x+5)}}}

and the final answer is

{{{(dy)/(dx) = ((x^2-8)^(1/3)(x^3+1)^(1/2)/(x^6-7x+5))*
(2x/(3(x-8))+3x^2/(2(x^3+1))-(6x^5-7)/(x^6-7x+5))}}}

Edwin</pre></font></b>