find the critical points of the following function. f(x)=x/(x^2 + 9) what is the derivative of f(x)=x/(x^2 +…

Explanation:

Step1: Apply quotient - rule

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

Step2: Simplify the expression

Expand the numerator: $1\cdot(x^{2}+9)-x\cdot(2x)=x^{2}+9 - 2x^{2}=9 - x^{2}$. So, $f'(x)=\frac{9 - x^{2}}{(x^{2}+9)^{2}}$.

Answer:

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