# How do you differentiate f(x) = (3 + 4x)/(1 + x^2)?

The quotient rule, which we'll use here, states that, be $y = f \frac{x}{g} \left(x\right)$, then $y ' = \frac{f ' \left(x\right) g \left(x\right) - f \left(x\right) g ' \left(x\right)}{g {\left(x\right)}^{2}}$.
$y ' = \frac{4 \cdot \left(1 + {x}^{2}\right) - \left(3 + 4 x\right) \cdot 2 x}{1 + {x}^{2}} ^ 2$
$y ' = \frac{4 + 4 {x}^{2} - 6 x - 8 {x}^{2}}{1 + {x}^{2}} ^ 2$
$y ' = \frac{- 4 {x}^{2} - 6 x + 4}{1 + {x}^{2}} ^ 2$