Logarithmic Derivative Exercise Question | Calculus

If $5^x+5^y=5^{x+y}$ then prove that ${\displaystyle \frac{dy}{dx}}+5^{y-x}=0$.

Differentiating with respect to $x$, we get

$5^x\ln (5)+5^y\ln (5){\displaystyle \frac{dy}{dx}}=5^{x+y}\ln (5)(1+{\displaystyle \frac{dy}{dx}})$
$\Rightarrow 5^x+5^y{\displaystyle \frac{dy}{dx}}=5^{x+y}(1+{\displaystyle \frac{dy}{dx}})$
$\Rightarrow 5^x+5^y{\displaystyle \frac{dy}{dx}}=5^{x+y}+5^{x+y}{\displaystyle \frac{dy}{dx}}$
$\Rightarrow {\displaystyle \frac{dy}{dx}}(5^{x+y}-5^y)+(5^{x+y}-5^x)=0$
$\Rightarrow {\displaystyle \frac{dy}{dx}}+{\displaystyle \frac{5^{x+y}-5^x}{5^{x+y}-5^y}}=0$
$\Rightarrow {\displaystyle \frac{dy}{dx}}+{\displaystyle \frac{5^y}{5^x}}=0$ $\Rightarrow {\displaystyle \frac{dy}{dx}}+5^{y-x}=0$