Question

Find dy/dx when y=x^(4lnx)

Answer 1

\[\large y=x^{4\ln x}=e^{\ln\left(x^{4\ln x}\right)}=e^{(4\ln x)\ln x}=e^{4(\ln x)^2}\]
Use the chain rule:
\[\large\begin{align*}\frac{dy}{dx}&=\frac{d}{dx}\left[e^{4(\ln x)^2}\right]\\\\
&=e^{4(\ln x)^2}\frac{d}{dx}\left[4(\ln x)^2\right]\\\\
&=x^{4\ln x}\frac{d}{dx}\left[4(\ln x)^2\right]\\\\
&=\cdots
\end{align*}\]