Derivative, quotient, and exponential rules (no calculator)

Question

Find dy/dx when e^y=x²/y.

Answer choices:

A. dy/dx = 2y/(xy-x)

B. dy/dx = (2y-xy)/x

C. dy/dx = 2y/(xy+x)

D. dy/dx = (2y+xy)/x

Answer 1

e^y = x²/y

ye^y = x²

ye^y(dy/dx) + e^y(dy/dx) = 2x

(dy/dx)(e^y)(y + 1) = 2x

(dy/dx)(x²/y)(y + 1) = 2x

(dy/dx)(x²y + x²) = 2xy

dy/dx = 2xy / [x²(y + 1)] = 2y / [x(y+1)] = 2y / (xy + x)