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Differentiation — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferentiation5 Marks
Question
Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$x^5 + y^5 = 5xy$
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Answer

We Heve, $x^5 + y^5 = 5xy$
Differentiating with respect to x, we get,
$\frac{\text{d}}{\text{dx}}\big(\text{x}^5\big)+\frac{\text{d}}{\text{dx}}\big(\text{y}^5\big)=\frac{\text{d}}{\text{dx}}\big(5\text{xy}\big)$
$\Rightarrow5\text{x}^4+5\text{y}^4\frac{\text{dy}}{\text{dx}}=5\Big[\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\frac{\text{dy}}{\text{dx}}\big(\text{x}\big)\Big]$
$\Rightarrow5\text{x}^4+5\text{y}^4\frac{\text{dy}}{\text{dx}}=5\Big[\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}\big(1\big)\Big]$
$\Rightarrow5\text{x}^4+5\text{y}^4\frac{\text{dy}}{\text{dx}}=5\text{x}\frac{\text{dy}}{\text{dx}}+5\text{y}$
$\Rightarrow5\text{y}^4\frac{\text{dy}}{\text{dx}}-5\text{x}\frac{\text{dy}}{\text{dx}}=5\text{y}-5\text{x}^4$
$\Rightarrow5\frac{\text{dy}}{\text{dx}}\big(\text{y}^4-\text{x}\big)=5\big(\text{y}-\text{x}^4\big)$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{5(\text{y}-\text{x}^4)}{5(\text{y}^4-\text{x})}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{y}-\text{x}^4}{\text{y}^4-\text{x}}$

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