Question
Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$(x + y)^2 = 2axy$
$(x + y)^2 = 2axy$
✓
Answer
We Have,$ (x + y)^2 = 2axy$
Differentiating with respect to x, we get,
$\Rightarrow\frac{\text{d}}{\text{dx}}\big(\text{x}+\text{y}\big)^2=\frac{\text{d}}{\text{dx}}\big(2\text{axy}\big)$
$\Rightarrow2(\text{x}+\text{y})\frac{\text{d}}{\text{dx}}(\text{x}+\text{y})=2\text{a}\Big[\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\frac{\text{d}}{\text{dx}}(\text{x})\Big]$
$\Rightarrow2(\text{x}+\text{y})\Big[1+\frac{\text{dy}}{\text{dx}}\Big]=2\text{a}\Big[\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}(1)\Big]$
$\Rightarrow2(\text{x}+\text{y})+2(\text{x}+\text{y})\frac{\text{dy}}{\text{dx}}=2\text{a}\text{x}\frac{\text{dy}}{\text{dx}}+2\text{ay}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}\big[2(\text{x}+\text{y})-2\text{a}\text{x}\big]=2\text{ay}-2(\text{x}+\text{y})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{2[\text{ay}-\text{x}-\text{y}]}{2[\text{x}+\text{y}-\text{a}\text{x}]}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\Big(\frac{\text{ay}-\text{x}-\text{y}}{\text{x}+\text{y}-\text{a}\text{x}}\Big)$
Differentiating with respect to x, we get,
$\Rightarrow\frac{\text{d}}{\text{dx}}\big(\text{x}+\text{y}\big)^2=\frac{\text{d}}{\text{dx}}\big(2\text{axy}\big)$
$\Rightarrow2(\text{x}+\text{y})\frac{\text{d}}{\text{dx}}(\text{x}+\text{y})=2\text{a}\Big[\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\frac{\text{d}}{\text{dx}}(\text{x})\Big]$
$\Rightarrow2(\text{x}+\text{y})\Big[1+\frac{\text{dy}}{\text{dx}}\Big]=2\text{a}\Big[\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}(1)\Big]$
$\Rightarrow2(\text{x}+\text{y})+2(\text{x}+\text{y})\frac{\text{dy}}{\text{dx}}=2\text{a}\text{x}\frac{\text{dy}}{\text{dx}}+2\text{ay}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}\big[2(\text{x}+\text{y})-2\text{a}\text{x}\big]=2\text{ay}-2(\text{x}+\text{y})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{2[\text{ay}-\text{x}-\text{y}]}{2[\text{x}+\text{y}-\text{a}\text{x}]}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\Big(\frac{\text{ay}-\text{x}-\text{y}}{\text{x}+\text{y}-\text{a}\text{x}}\Big)$
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