Question

Please help with a derivative

Answer 1

\[(3\sin ^{-1}(5x))^{x}\]

Answer 2

Let \(y=(3\sin^{-1}(5x))^x\), then \(\ln y=x \ln(3\sin^{-1}(5x))=x\ln3+x\ln(\sin^{-1}(5x))\).

Differentiate both sides w.r.t. \(x\):
\[\begin{align*}\frac{1}{y}\frac{dy}{dx}&=\frac{d}{dx}[x\ln3]+\frac{d}{dx}\left[x\ln(\sin^{-1}(5x))\right]\\\\
&=\ln3+\frac{d}{dx}[x]\ln(\sin^{-1}(5x))+x\frac{d}{dx}\left[\ln(\sin^{-1}(5x))\right]\\\\
&=\ln3+\ln(\sin^{-1}(5x))+x\frac{1}{\sin^{-1}(5x)}\frac{d}{dx}\left[\sin^{-1}(5x)\right]\\\\
&=\ln3+\ln(\sin^{-1}(5x))+\frac{x}{\sin^{-1}(5x)}\frac{5}{\sqrt{1-25x^2}}\frac{d}{dx}[5x]\\\\
&=\ln3+\ln(\sin^{-1}(5x))+\frac{x}{\sin^{-1}(5x)}\frac{25}{\sqrt{1-25x^2}}
\end{align*}\]
Now solve for \(\dfrac{dy}{dx}\).