Answer:
E. dy/dx = y (y − x ln y) / (x (x − y ln x))
Step-by-step explanation:
yˣ = xʸ
Take log of both sides.
ln yˣ = ln xʸ
x ln y = y ln x
Implicit derivative (use product rule and chain rule).
x (1/y dy/dx) + ln y (1) = y (1/x) + ln x (dy/dx)
Solve for dy/dx.
x/y dy/dx + ln y = y/x + ln x dy/dx
x² dy/dx + xy ln y = y² + xy ln x dy/dx
(x² − xy ln x) dy/dx = y² − xy ln y
dy/dx = (y² − xy ln y) / (x² − xy ln x)
dy/dx = y (y − x ln y) / (x (x − y ln x))
