Logarithmic Derivative Exercise Question | Calculus

If \(5^x + 5^y = 5^{x + y}\) then prove that \(\dfrac{dy}{dx} + 5^{y - x} = 0\).

Differentiating with respect to \(x\), we get

\(5^x\ln(5)+5^y\ln(5)\dfrac{dy}{dx}=5^{x+y}\ln(5)(1+\dfrac{dy}{dx})\)
\(\Rightarrow 5^x+5^y\dfrac{dy}{dx}=5^{x+y}(1+\dfrac{dy}{dx})\)
\(\Rightarrow 5^x+5^y\dfrac{dy}{dx}=5^{x+y}+5^{x+y}\dfrac{dy}{dx}\)
\(\Rightarrow \dfrac{dy}{dx}(5^{x+y}−5^y)+(5^{x+y}−5^x)=0\)
\(\Rightarrow \dfrac{dy}{dx}+\dfrac{5^{x+y}−5^x}{5^{x+y}−5^y}=0\)
\(\Rightarrow \dfrac{dy}{dx}+\dfrac{5^y}{5^x}=0\) \(\Rightarrow \dfrac{dy}{dx}+5^{y−x}=0\)