How do I find the derivative of this function?

You won't be needing the use of natural logs, just the chain rule, product rule, and a little implicit differentiation.

(3x^5 * e^8y) + (5y^3 * e^5x) = 18

d/dx => 0 =
[(15x^4)(e^8y) + (e^8y)(8)(y')(3x^5)] + [(15y^2)(y')(e^5x) + (e^5x)(5)(5y^3)]

Whew....that doesn't all fit on one line

Now that we have that, it is best to simplify to make it easier to handle.

d/dx => 0 = [3e^8y(5x^4) + (8x^5)(y')] + [5e^5x(3y^2)(y') + (5y^3)]

Rearrange the equation so that you can see the easiest way to factor out the (y')s.  Note that all the terms are being added, so you can omit the square parentheses on each part.

d/dx => 0 = 5e^5x(3y^2)(y') + (8x^5)(y') + 3e^8y(5x^4) + (5y^3)

Now factor out y'.

d/dx => 0 = y'[(5e^5x(3y^2) + (8x^5)] + 3e^8y(5x^4) + (5y^3))

Subtract.

d/dx => -(3e^8y(5x^4) + (5y^3)) = y'[(5e^5x(3y^2) + (8x^5)]

Divide to isolate y'.

d/dx => y'= -3e^8y(5x^4) + (5y^3))
.................------- --------- -------- -------
...................5e^5x(3y^2) + (8x^5)

The line between there represents my style of division .

If you wish to, simplify further.