Given that w=x * e^-w use implicit differentiation to show that dw/dx=1/(e^w + x)

Given that w=xe^-w use implicit differentiation to show that dw/dx = 1/(e^w+x)Answer:Use product rule to simplify:dw/dx = x(de^-w/dx) + e^-w(dx/dx)Use chain rule to simplify even further:dw/dx = -xe^-w(dw/dx) + e^-wWe know from the original formula that w = xe^-w. Therefore, replace:dw/dx = -w*(dw/dx) + e^-wRe-arrange to isolate the derivative:(dw/dx)(1+w) = e^-wdw/dx = (e^-w)/(1+w)Re-arrange to achieve form asked for, knowing that x = we^w from original formula given:dw/dx = 1/(e^w+ we^w)dw/dx = 1/(e^w+x)q.e.d.