Question

Find dy/dx if e^x+y=xy

Answer 1

Given that xy = e^x+y

Taking log both sides, we get

log_exy = x+y (Since log_ab^c = c log_ab)

Since log_abc = log_ab + log_ac, we get

log_ex + log_ey = x+y

Differentiating with respect to x, we get

1/x + 1/y dy/dx = 1 + dy/dx

Or

\(\frac{dy}{dx}(\frac{y-1}{y})=\frac{1-x}{x}\)

Hence, \(\frac{dy}{dx}=\frac{y(1-x)}{x(y-1)}\)