Question

Here's one: find the derivative of (xy)^x = e
Thanks!

Answer 1

This is what I did:
I took the natural log to both sides
xln(xy)=1
I divided x out of each side
ln(xy)=1/x
Used properties of logarithms to separate out ln(xy)
lnx + lny=1/x
lny=-lnx+1/x
then I took the derivative to both sides
y'/y=-[(1/x^2)+(1/x)]
I then multiplied both sides by y, which I just found by using the earlier equation: lny=-lnx+1/x
y=e^(-lnx+1/x)

y'=-[(1/x^2)+(1/x)][e^(-lnx+1/x)]

But I'm sure there's an easier or different way, I just went about it this route since this is where I felt most comfortable.

Answer 2

Very interesting. I asked the question somewhere else and got a different approach but same response. This is actually very helpful to see it done a couple different ways. And thanks for the extra explanations!