Question

If x^y = e^{(x – y)} , then \(\frac {dy}{dx}\) =?

• A. \(\frac {log \ x}{(1+ log x)^2}\)
• B. \(\frac {log \ x}{(1+ log x)}\)
• C. \(\frac {xlog \ x}{(1+ log x)^2}\)
• D. \(\frac {log \ x}{x(1+ log x)^2}\)

Answer 1

The Correct Option is A

x^y = e^(x−y)
On taking log both sides:
logx^y = log e^(x−y).
ylogx = (x−y)log e
ylogx = x−y
y+ylogx = x;
y = \(\frac {x}{1+logx}\)
On differentiating both sides with respect to x:
\(\frac {dy}{dx}\) = \(\frac {(1+log \ x)1- x(\frac{1}{x})}{(1+ log x)^2}\) 
\(\frac {dy}{dx}\) = \(\frac {1+log \ x-1}{(1+ log x)^2}\)
\(\frac {dy}{dx}\) = \(\frac {log \ x}{(1+ log x)^2}\)
Therefore, the correct option is (A) \(\frac {log \ x}{(1+ log x)^2}\).