e(2x+3y) = x2 - ln(xy3)
Nicolee, the first thing I would do would be to expand the "ln" using logarithm properties to keep from doing a complicated chain rule.
e(2x+3y) = x2 - (lnx + 3lny)
e(2x+3y) = x2 - lnx -3lny
No differentiate each side
deriv of eanything is eanything * deriv of "anything". So the left side becomes
e(2x+3y)*[2 + 3 (dy/dx)]
diff the right side and get 2x - (1/x) - 3(1/y)(dy/dx)
distribute on the left
e(2x+3y)(2) + e(2x+3y)[3(dy/dx)] = 2x - (1/x) -(3/y)(dy/dx)
Now put all the (dy/dx)'s on the left. Put everything else on the right.
Factor a (dy/dx) out of every term on the left
And divide both sides by the stuff on the left in the parentheses.
Sorry - gotta run or I would show these steps...