<s>[INST] Differentiate and simplify. $m(x) = \frac{x}{\sqrt{4x-3}}$ My work so far is:
\begin{align}
m'(x) &= \frac{(1)(\sqrt{4x-3})-(x)(1/2)(4x-3)^{-1/2}(4)}{(\sqrt{4x-3})^2} \\
&= \frac{\sqrt{4x-3} - 2x(4x-3)^{1/2}}{4x-3}
\end{align}
and now I'm stuck on how to simplify further
 [/INST] Another way which makes life easier when you face products, quotients, powers : logarithmic differentiation $$m=\frac{x}{\sqrt{4x-3}}\implies \log(m)=\log(x)-\frac 12 \log(4x-3)$$ Differentiate $$\frac{m'} m=\frac 1x-\frac 12 \times \frac 4{4x-3}=\frac{2x-3}{x(4x-3)}$$ $$m'=m\times \frac{2x-3}{x(4x-3)}=\frac{x}{\sqrt{4x-3}}\times \frac{2x-3}{x(4x-3)}=\frac{2x-3}{(4x-3)^{3/2}}$$
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