Question

If x^y = e^{x – y}, prove that `dy/dx = log x/((1 + log x)^2`.

Answer 1

Given: x^y = e^{x – y}

To Prove: `dy/dx = log x/((1 + log x)^2`

Proof:

Taking log on both sides

log_e (x^y) = log_e (e^{x – y})

⇒ y log x = x – y

⇒ y log x + y = x

⇒ `y = x/(1 + log x)`

Now differentiate w.r.t. x

`dy/dx = ((1 + log x) d/dx x - x d/dx (1 + log x))/(1 + log x)^2`

⇒ `dy/dx = ((1 + log x) - x(1/x))/(1 + log x)^2`

⇒ `dy/dx = log x/(1 + log x)^2`

Hence Proved.