If (cos x)^y= ( y)^x, find dy dx . - Vidyadip

Question

If $(cos x)^y= (\sin y)^x$, find $\frac{\text{dy}}{\text{dx}}.$

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Answer

$(\cos x)^y = (\sin y)^x$
$\Rightarrow$ y log cos x = x log sin y
$\therefore$ log (cos x). $\frac{\text{dy}}{\text{dx}}$ - y . tan x = log sin y + x cot y. $\frac{\text{dy}}{\text{dx}}$
$\therefore$$\frac{\text{dy}}{\text{dx}}$ (log cos x - x cot y) = log sin y + y tan x
$\therefore$$\frac{\text{dy}}{\text{dx}}=\frac{\log\sin\text{y + y tan x}}{\text{log cos x - x}\cdot\text{cot y}}.$

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