If ( x)^y=( y)^x, then d y d x is equal to: - Vidyadip

MCQ

If $(\cos x)^y=(\cos y)^x$, then $\frac{d y}{d x}$ is equal to:

A
$\frac{y \tan x+\log (\cos y)}{x \tan y-\log (\cos x)}$
B
$\frac{x \tan y+\log (\cos x)}{y \tan x+\log (\cos y)}$
C
$\frac{y \tan x-\log (\cos y)}{x \tan y-\log (\cos x)}$
D
$\frac{y \tan x+\log (\cos y)}{x \tan y+\log (\cos x)}$

✓

Answer

Given, $(\cos x)^y=(\cos y)^x$
Taking log on both sides, we get
$\log \left[(\cos x)^y\right]=\log \left[(\cos y)^x\right] \Rightarrow y \log (\cos x)=x \log (\cos y)$
Differentiate w.r.t. $x$, we get
$\begin{array}{l}
\frac{d y}{d x} \log (\cos x)+\frac{y}{\cos x}(-\sin x)=\log (\cos y)+\frac{x}{\cos y}(-\sin y) \cdot \frac{d y}{d x} \\
\Rightarrow \frac{d y}{d x}(\log (\cos x)+x \tan y)=\log (\cos y)+y \tan x \\
\Rightarrow \frac{d y}{d x}=\frac{y \tan x+\log (\cos y)}{x \tan y+\log (\cos x)}
\end{array}
$

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