# 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}$