Question

Differentiate
If x^y – y^x = a^b, find $\frac{dy}{dx}.$

Answer 1

$x^y-y^x=a^b$

$\Rightarrow e^{y\ln x}-e^{x\ln y}=a^b\left[e^{\ln f\left(x\right)}=f\left(x\right)\right]$

Differentiating both sides with respect to x, we get

$\frac{d}{dx}{e}^{y\ln x}-\frac{d}{dx}{e}^{x\ln y}=\frac{d}{dx}{a}^b$

$\Rightarrow e^{y\ln x}\left(y\times \frac{d}{dx}\ln x+\ln x\times \frac{dy}{dx}\right)-e^{x\ln y}\left(x\times \frac{d}{dx}\ln y+\ln y\times \frac{d}{dx}x\right)=0$

$\Rightarrow x^y\left(y\times \frac{1}{x}+\ln x\frac{dy}{dx}\right)-y^x\left(x\times \frac{1}{y}\frac{dy}{dx}+\ln y\right)=0$

$\Rightarrow x^{y-1}y+x^y\ln x\frac{dy}{dx}-x{y}^{x-1}\frac{dy}{dx}-y^x\ln y=0$

$\Rightarrow \left(x^y\ln x-x{y}^{x-1}\right)\frac{dy}{dx}=y^x\ln y-x^{y-1}y$

$\Rightarrow \frac{dy}{dx}=\frac{y\left(y^{x-1}\ln y-x^{y-1}\right)}{x\left(x^{y-1}\ln x-y^{x-1}\right)}$