Question

If xy = 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

Answer: (A)

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

Answer 2

xy = e(x−y)
On taking log both sides:
logxy = 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) $\frac {log \ x}{(1+ log x)^2}$.