Question

find the derivative of the function $f(x)$, below. it may be to your advantage to simplify first.
$f(x)=\frac{5 + x}{6 + 9x+2x^{2}}$
$f(x)=$

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = 5 + x$, so $u'=1$; $v=6 + 9x+2x^{2}$, so $v'=9 + 4x$.

Step2: Substitute into quotient - rule formula

$f'(x)=\frac{(1)\times(6 + 9x+2x^{2})-(5 + x)\times(9 + 4x)}{(6 + 9x+2x^{2})^{2}}$.

Step3: Expand the numerator

Expand $(6 + 9x+2x^{2})-(45 + 20x+9x+4x^{2})$.
$6 + 9x+2x^{2}-45 - 29x - 4x^{2}=-2x^{2}-20x - 39$.

Answer:

$\frac{-2x^{2}-20x - 39}{(6 + 9x+2x^{2})^{2}}$