Question

If x^y = e^x–y, then $\frac{dy}{dx}$ is equal to

• A. $\frac{y}{(1 + \log x)^{2}}$
• B. $\frac{x}{(1 + \log x)^{2}}$
• C. $\frac{\log x}{(1 + \log x)^{2}}$
• D. none of these

Answer 1

Answer: (C)

$\frac{\log x}{(1 + \log x)^{2}}$

Answer 2

x^y = e^{x – y} ⇒ y log x = (x – y) . 1 ⇒ y(1 + log x) = x

y = $\frac{x}{1 + \log x}$ ⇒ $\frac{dy}{dx} = \frac{(1 + \log x).1 - x(0 + 1/x)}{(1 + \log x)^{2}}$

= logx / (1 + log x)²