Question

If $x^y=a^x.Provethat\frac{dy}{dx}=\frac{x{\log }_{e}a-y}{x{\log }_{e}x}$

Answer 1

$x^y=a^x$

taking logarith on both sides , we get

$y$ log $x=x$ log $a$

differentiating both sides with respect to $x$

$\frac{dy}{dx}$ log $x+y.\frac{1}{x}=$ log $a$

$\Rightarrow \frac{dy}{dx}\times x$ log $x+y=x$ log $a$

$\Rightarrow \frac{dy}{dx}=\frac{xloga-y}{xlogx}$