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/(ew+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 = wew from original formula given:dw/dx = 1/(ew+ wew)dw/dx = 1/(ew+x)q.e.d.
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