Question 1187768
<pre>
I think it was meant to be this:</pre>Use logarithmic differentiation to find the derivative of the function.<pre>

{{{matrix(2,3,"","","",y,""="",x^(-8/x))}}}

The hard part is going to be the algebra of simplifying the derivative:

{{{matrix(2,3,"","","",y,""="",x^((-8x^(-1))))}}}

Taking natural logs of both sides:

{{{matrix(2,3,"","","",ln(y),""="",ln(x^((-8x^(-1)))))}}}

Using a rule of logarithms:

{{{matrix(2,3,"","","",ln(y),""="",-8x^(-1)ln(x))}}}

Taking derivatives of both sides

{{{"y'"/y}}}{{{""=""}}}{{{-8x^(-1)*(1/x) + ln(x)*8x^(-2)}}}

{{{"y'"/y}}}{{{""=""}}}{{{-8x^(-1)*(x^(-1)) + ln(x)*8x^(-2)}}}

{{{"y'"/y}}}{{{""=""}}}{{{-8x^(-2) + ln(x)*8x^(-2)}}}

{{{"y'"/y}}}{{{""=""}}}{{{ ln(x)*8x^(-2)-8x^(-2)}}}

{{{"y'"/y}}}{{{""=""}}}{{{8x^(-2)(ln(x)-1^"")}}}

Multiply both sides by y

{{{"y'"}}}{{{""=""}}}{{{8yx^(-2)(ln(x)-1^"")}}}

Substitute {{{matrix(2,3,"","","",y,""="",x^((-8x^(-1))))}}}

{{{matrix(2,3,"","","","y'",""="",8(x^((-8x^(-1))))x^(-2)(ln(x)-1^""))}}}

{{{matrix(2,3,"","","","y'",""="",8(x^((-8x^(-1)-2)))(ln(x)-1^""))}}}

{{{matrix(2,3,"","","","y'",""="",8(x^(-2(4x^(-1)+1)))(ln(x)-1^""))}}}

{{{matrix(2,3,"","","","y'",""="",8(x^(-2(4/x+1)))(ln(x)-1^""))}}}

{{{matrix(2,3,"","","","y'",""="",8(x^(-2(4/x+x/x)))(ln(x)-1^""))}}}

{{{matrix(2,3,"","","","y'",""="",8(x^(-2((4+x)/x)))(ln(x)-1^""))}}}

{{{matrix(2,3,"","","","y'",""="",8(x^(-2((x+4)/x)))(ln(x)-1^""))}}}

Whew! (worn out!)  <font size=5><font face="wingdings">J</font><pre></font>
Edwin</pre>