The equation is x^y * y^x = 1. \ \

Differentiate the above equation with respect to x. \ \

[ x^y * y^x ] \\'  = 1\\' \ \

Use product rule differentiation formula : (uv) \\' = uv \\' + vu \\'. \ \

Derivative of constant is zero.

(x^y)(y^x) \\'  + (y^x)(x^y) \\' = 0

Let y^x = m. \ \

Apply logarithm on each side.

ln y^x = ln m

Apply power property of logarithm : log_a(m)ⁿ = nlog_a(m).

x ln y = ln m

Apply derivative on each side.

(x ln y) \\' = (ln m) \\'

x (ln y) \\' + (ln y)(x) \\' = (ln m) \\'

Use the formula : (ln x) = 1/x.

x (1/y)y\\' + (ln y)(1) = (1/m)m\\'

(xy \\' / y) + ln y = (1/y^x)(y^x) \\'

(y^x) \\' = [y^x(xy \\' + y ln y)] / y