# Derivative of Rational Functions
For functions $u$ and $v$ of $x$, the derivative of $\frac u v$ is:
$\frac d {dx} \left[ \frac u v \right] = \frac {du\cdot v - u\cdot dv} {v^2}$ ^mmdg5g
Recall the product and reciprocal rules:
![[Derivative Product Rule#^a1ucyf]]
![[Derivative Chain Rule#^1vw3jd]]
$
\begin{align}
\frac d {dx} \left[ \frac u v \right]
&= \frac d {dx} \left[ u \cdot \frac 1 v \right] \\
&= \frac {du} {v} + u (-v^{-2})dv \\
&= \frac {du \cdot v} {v^2} + \frac {-u \cdot dv} {v^2}\\
&= \frac {du \cdot v - u \cdot dv} {v^2}
\end{align}$