# How do you differentiate f(x)=(5x-2)/(x^2+1) using the quotient rule?

Mar 15, 2018

See explanation.

#### Explanation:

The Quotient Rule says that

${\left[f \frac{x}{g} \left(x\right)\right]}^{'} = \frac{{f}^{'} \left(x\right) \cdot g \left(x\right) - f \left(x\right) {g}^{'} \left(x\right)}{{\left[g \left(x\right)\right]}^{2}}$

In the given example we have:

$f \left(x\right) = 5 x - 2$

$f ' \left(x\right) = 5$

$g \left(x\right) = {x}^{2} + 1$

$g ' \left(x\right) = 2 x$

So the derivative is:

${\left[\frac{5 x - 2}{{x}^{2} + 1}\right]}^{'} = \frac{5 \left({x}^{2} + 1\right) - \left(5 x - 2\right) \cdot \left(2 x\right)}{{x}^{2} + 1} ^ 2$

${\left[\frac{5 x - 2}{{x}^{2} + 1}\right]}^{'} = \frac{5 {x}^{2} + 5 - 10 {x}^{2} + 4 x}{{x}^{2} + 1} ^ 2$

${\left[\frac{5 x - 2}{{x}^{2} + 1}\right]}^{'} = \frac{- 5 {x}^{2} + 4 x + 5}{{x}^{2} + 1} ^ 2$