Question

If $x^y+y^x={\left(x+y\right)}^{x+y},\mathrm{find}\frac{dy}{dx}$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have},x^y+y^x={\left(x+y\right)}^{x+y} \\ \Rightarrow e^{\log x^y}+e^{\log y^x}=e^{\log {\left(x+y\right)}^{\left(x+y\right)}} \\ \Rightarrow e^{y\log x}+e^{x\log y}=e^{\left(x+y\right)\log \left(x+y\right)} \end{gathered}$$

Differentiating with respect to x using chain rule and product rule,
$$\begin{gathered} \Rightarrow \frac{d}{dx}\left(e^{y\log x}\right)+\frac{d}{dx}\left(e^{x\log y}\right)=\frac{d}{dx}{e}^{\left(x+y\right)\log \left(x+y\right)} \\ \Rightarrow e^{y\log x}\left[y\frac{d}{dx}\left(\log x\right)+\log x\frac{dy}{dx}\right]+e^{x\log y}\left[x\frac{d}{dx}\log y+\log y\frac{d}{dx}\left(x\right)\right]=e^{\left(x+y\right)\log \left(x+y\right)}\frac{d}{dx}\left[\left(x+y\right)\log \left(x+y\right)\right] \\ \Rightarrow e^{\log x^y}\left[y\left(\frac{1}{x}\right)+\log x\frac{dy}{dx}\right]+e^{\log x}\left[\frac{x}{y}\frac{dy}{dx}+\log y\left(1\right)\right]=e^{\log {\left(x+y\right)}^{\left(x+y\right)}}\left[\left(x+y\right)\frac{d}{dx}\log \left(x+y\right)+\log \left(x+y\right)\frac{d}{dx}\left(x+y\right)\right] \\ \Rightarrow x^y\left[\frac{y}{x}+\log x\frac{dy}{dx}\right]+y^x\left[\frac{x}{y}\frac{dy}{dx}+\log y\right]={\left(x+y\right)}^{\left(x+y\right)}\left[\left(x+y\right)\frac{1}{\left(x+y\right)}\frac{d}{dx}\left(x+y\right)+\log \left(x+y\right)\left(1+\frac{dy}{dx}\right)\right] \\ \Rightarrow x^y\times \frac{y}{x}+x^y\log x\frac{dy}{dx}+y^x\times \frac{x}{y}\frac{dy}{dx}+y^x\log y={\left(x+y\right)}^{\left(x+y\right)}\left[1\times \left(1+\frac{dy}{dx}\right)+\log \left(x+y\right)\left(1+\frac{dy}{dx}\right)\right] \\ \Rightarrow x^{y-1}\times y+x^y\log x\frac{dy}{dx}+y^{x-1}\times x\frac{dy}{dx}+y^x\log y={\left(x+y\right)}^{\left(x+y\right)}+{\left(x+y\right)}^{\left(x+y\right)}\frac{dy}{dx}+{\left(x+y\right)}^{\left(x+y\right)}\log \left(x+y\right)+{\left(x+y\right)}^{\left(x+y\right)}\log \left(x+y\right)\frac{dy}{dx} \\ \Rightarrow \frac{dy}{dx}\left[x^y\log x+x{y}^{x-1}-{\left(x+y\right)}^{\left(x+y\right)}\left\{1+\log \left(x+y\right)\right\}\right]={\left(x+y\right)}^{\left(x+y\right)}\left\{1+\log \left(x+y\right)\right\}-x^{y-1}\times y-y^x\log y \\ \Rightarrow \frac{dy}{dx}=\left[\frac{{\left(x+y\right)}^{\left(x+y\right)}\left\{1+\log \left(x+y\right)\right\}-y{x}^{y-1}-y^x\log y}{x^y\log x+x{y}^{x-1}-{\left(x+y\right)}^{\left(x+y\right)}\left\{1+\log \left(x+y\right)\right\}}\right] \end{gathered}$$