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Question
Find \[\frac{dy}{dx}\] in the following case \[x^5 + y^5 = 5 xy\] ?
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Solution
\[\text{We have, }x^5 + y^5 = 5xy\]
Differentiating with respect to x, we get,
\[\frac{d}{dx}\left( x^5 \right) + \frac{d}{dx}\left( y^5 \right) = \frac{d}{dx}\left( 5xy \right)\]
\[ \Rightarrow 5 x^4 + 5 y^4 \frac{d y}{d x} = 5\left[ x\frac{d y}{d x} + y\frac{d}{d x}\left( x \right) \right]\]
\[ \Rightarrow 5 x^4 + 5 y^4 \frac{d y}{d x} = 5\left[ x\frac{d y}{d x} + y\left( 1 \right) \right]\]
\[ \Rightarrow 5 x^4 + 5 y^4 \frac{d y}{d x} = 5x\frac{d y}{d x} + 5y\]
\[ \Rightarrow 5 y^4 \frac{d y}{d x} - 5x\frac{d y}{d x} = 5y - 5 x^4 \]
\[ \Rightarrow 5\frac{d y}{d x}\left( y^4 - x \right) = 5\left( y - x^4 \right)\]
\[ \Rightarrow \frac{d y}{d x} = \frac{5\left( y - x^4 \right)}{5\left( y^4 - x \right)}\]
\[ \Rightarrow \frac{d y}{d x} = \frac{y - x^4}{y^4 - x}\]
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