Given: #(sin(pix)+cos(piy))^2 = 2#
Differentiate:
#(d((sin(pix)+cos(piy))^2))/dx = (d(2))/dx#
Use the chain rule on #(d((sin(pix)+cos(piy))^2))/dx#
The chain rule is:
#(d(g(h(x,y))))/dx = (dg)/(dh)(dh)/dx#
let #h(x,y) = sin(pix)+cos(piy)#, then:
#(dh)/dx = picos(pix) -pisin(piy)dy/dx#
#g = (h(x,y))^2#
#(dg)/(dh) = 2h(x,y)#
#(d((sin(pix)+cos(piy))^2))/dx = 2h(x,y)(picos(pix) -pisin(piy)dy/dx)#
#(d((sin(pix)+cos(piy))^2))/dx = 2(sin(pix)+cos(piy))(picos(pix) -pisin(piy)dy/dx)#
The right side is just derivative of a constant:
#(d(2))/dx = 0#
Put the terms back into the equation:
#2(sin(pix)+cos(piy))(picos(pix) -pisin(piy)dy/dx) = 0#
Solve for #dy/dx#
#-cos(pix) + sin(piy)dy/dx = 0#
#sin(piy)dy/dx = cos(pix)#
#dy/dx = cos(pix)/sin(piy)#
