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