<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] In general we have $\frac{\partial}{\partial x} \frac{f(x)}{g(x)}= \frac{\frac{\partial f(x)}{\partial x} \times g(x) - \frac{\partial g(x)}{\partial x} \times f(x)}{g^2(x)}$. So in the case of your question, set $f(x)= x$ and $g(x) = (4x - 3)^{\frac{1}{2}}$. And we have:
$\frac{\partial}{\partial x} (x) = 1$ and
$\frac{\partial}{\partial x} ((4x - 3)^{\frac{1}{2}}) = \frac{1}{2} \times 4  \times (4x - 3)^{- \frac{1}{2}}= 2 \times (4x - 3)^{- \frac{1}{2}}$. So we have:
$\frac{\partial}{\partial x} \frac{x}{(4x - 3)^{\frac{1}{2}}}= \frac{1 \times (4x - 3)^{\frac{1}{2}} - 2 \times (4x - 3)^{- \frac{1}{2}} \times x }{4x - 3} = \frac{2x-3}{(4x - 3)^{\frac{3}{2}}}$.
And we are done.
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