Question

If $x^y \cdot y^x=16$ then $\left(\frac{d y}{d x}\right)_{(2,2)}$ is equal to

• A. 1
• B. -1
• C. 0
• D. $\ln 2+1$

Answer 1

Answer: (B)

Taking log on both sides,
$y \ln x + x \ln y =\ln 16$
differentiating w.r.t. $x$, we get
$\frac{y}{x}+\ln x \cdot y^{\prime}+\frac{x}{y} \cdot y^{\prime}+\ln y=0 $
$\text { putting } x=y=2 $
$1+\ln 2 \cdot y^{\prime}+y^{\prime}+\ln 2=0 $
$y^{\prime}=-1 $