SOLUTION: Use logarithmic differentiation to find the derivative of the function. y=x^−8/x y′=

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Question 1187768: Use logarithmic differentiation to find the derivative of the function.
y=x^−8/x
y′=

Found 2 solutions by greenestamps, AnlytcPhil:
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


Not clear....

Is it y=%28x%5E%28-8%29%29%2Fx ?

Or is it y=%28x%5E%28-8%2Fx%29%29

Or maybe even something different still....

Re-post

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
I think it was meant to be this:
Use logarithmic differentiation to find the derivative of the function.

matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2Cx%5E%28-8%2Fx%29%29

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



Taking natural logs of both sides:



Using a rule of logarithms:



Taking derivatives of both sides

%22y%27%22%2Fy%22%22=%22%22-8x%5E%28-1%29%2A%281%2Fx%29+%2B+ln%28x%29%2A8x%5E%28-2%29

%22y%27%22%2Fy%22%22=%22%22-8x%5E%28-1%29%2A%28x%5E%28-1%29%29+%2B+ln%28x%29%2A8x%5E%28-2%29

%22y%27%22%2Fy%22%22=%22%22-8x%5E%28-2%29+%2B+ln%28x%29%2A8x%5E%28-2%29

%22y%27%22%2Fy%22%22=%22%22+ln%28x%29%2A8x%5E%28-2%29-8x%5E%28-2%29

%22y%27%22%2Fy%22%22=%22%228x%5E%28-2%29%28ln%28x%29-1%5E%22%22%29

Multiply both sides by y

%22y%27%22%22%22=%22%228yx%5E%28-2%29%28ln%28x%29-1%5E%22%22%29

Substitute 















Whew! (worn out!)  J

Edwin