Question

Find $\frac{dy}{dx}$ in each of the following cases:

$x^5+y^5=5xy$

Answer 1

$\mathrm{We}\mathrm{have},x^5+y^5=5xy$
Differentiating with respect to x, we get,

$$\begin{gathered} \frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(y^5\right)=\frac{d}{dx}\left(5xy\right) \\ \Rightarrow 5x^4+5y^4\frac{\mathit{d}y}{\mathit{d}x}=5\left[x\frac{\mathit{d}y}{\mathit{d}x}+y\frac{\mathit{d}}{\mathit{d}x}\left(x\right)\right] \\ \Rightarrow 5x^4+5y^4\frac{\mathit{d}y}{\mathit{d}x}=5\left[x\frac{\mathit{d}y}{\mathit{d}x}+y\left(1\right)\right] \\ \Rightarrow 5x^4+5y^4\frac{\mathit{d}y}{\mathit{d}x}=5x\frac{\mathit{d}y}{\mathit{d}x}+5y \\ \Rightarrow 5y^4\frac{\mathit{d}y}{\mathit{d}x}-5x\frac{\mathit{d}y}{\mathit{d}x}=5y-5x^4 \\ \Rightarrow 5\frac{\mathit{d}y}{\mathit{d}x}\left(y^4-x\right)=5\left(y-x^4\right) \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}=\frac{5\left(y-x^4\right)}{5\left(y^4-x\right)} \\ \Rightarrow \frac{\mathit{d}y}{\mathit{d}x}=\frac{y-x^4}{y^4-x} \end{gathered}$$