Question

Use implicit differentiation to show

dy/dx = [(1-xy)cot(y)]/[x^2 cot(y) + xcsc^2(y)]

if xy = ln(x cot y).

Answer 1

from xy = ln(x coty)
x dy/dx + y = (x(-csc^y dy/dx) + coty)/(x coty)
cross-multiply
(x^2)(coty)dy/dx + xycoty = -x(csc^2y)dy/dx + coty
(x^2)(coty)dy/dx + x(csc^2y)dy/dx = coty - xycoty
dy/dx((x^2)(coty) + x(csc^2y)) = coty(1 - xy)
dy/dx = (1-xy)coty/[(x^2)(coty)dy/dx + x(csc^2y)]