If y=x , prove that dy dx = y x(1-x ) - Vidyadip

Question

If $\text{y}=\text{x}\sin\text{y},$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}(1-\text{x}\cos\text{y})}$

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Answer

We have, $\text{y}=\text{x}\sin\text{y}\ .....(\text{i})$
Differentiating with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}(\text{x}\sin\text{y})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{x}\frac{\text{d}}{\text{dx}}(\sin\text{y})+\sin\text{y}\frac{\text{d}}{\text{dx}}(\text{x})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{x}\cos\text{y}\frac{\text{dy}}{\text{dx}}+\sin\text{y}(1)$
$\Rightarrow\frac{\text{dy}}{\text{dx}}-\text{x}\cos\text{y}\frac{\text{dy}}{\text{dx}}=\sin\text{y}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}(1-\text{x}\cos\text{y})=\sin\text{y}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\sin\text{y}}{(1-\text{x}\cos\text{y})}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}(1-\text{x}\cos\text{y})}\Big[\because\sin\text{y}=\frac{\text{y}}{\text{x}}\Big]$

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