Question

If $x^y=3^{x-y}$ then $\frac{dy}{dx}=$ ?

• A. $\frac{x\log 3+y}{x\log 3x}$
• B. $\frac{y-x\log x}{x\log 3}$
• C. $\frac{x\log 3-y}{x\log 3x}$
• D. $\frac{y-x\log 3}{x\log 3x}$

Answer 1

The correct option is C $\frac{x\log 3-y}{x\log 3x}$

$\because x^y=3^{x-y}\Rightarrow \log x^y=\log 3^{x-y}\Rightarrow y\log x=(x-y)\log 3\Rightarrow y\frac{d}{dx}\left(\log x\right)+\log x\frac{dy}{dx}=(1-\frac{dy}{dx})\log 3\Rightarrow \frac{y}{x}+\log x\frac{dy}{dx}=\log 3-\frac{dy}{dx}\log 3\Rightarrow \frac{dy}{dx}(\log x+\log 3)=\log 3-\frac{y}{x}\Rightarrow \frac{dy}{dx}\log 3x=\frac{x\log 3-y}{x}\Rightarrow \frac{dy}{dx}=\frac{x\log 3-y}{x\log 3x}$