Can someone give me a hint about transforming $\displaystyle x^{4x}$ into a quotient? I think it involves setting y equal to it and taking a log or something.
A quotient? What about $\displaystyle \dfrac{1}{x^{-4x}}$?
But I suspect you are really talking about the derivative. If we let $\displaystyle y= x^{4x}$ then $\displaystyle ln(y)= ln(x^{4x})= 4x ln(x)$ so that $\displaystyle \frac{1}{y} y'= 4 ln(x)+ \frac{4x}{x}= 4 ln(x)+ 4$ so that $\displaystyle y'= 4y(ln(x)+ 1)= 4x^4(ln(x)+ 1)$.