differentiate. f(t) = 6t / (9 + t^2) f(t) =

Explanation:

Step1: Recall quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = 6t$, $v=9 + t^{2}$.

Step2: Find $u'$ and $v'$

Differentiate $u = 6t$ with respect to $t$, we get $u'=6$. Differentiate $v = 9 + t^{2}$ with respect to $t$, we get $v' = 2t$.

Step3: Apply quotient - rule

Substitute $u$, $u'$, $v$, $v'$ into the quotient - rule formula: [ \begin{align*} f'(t)&=\frac{6(9 + t^{2})-6t(2t)}{(9 + t^{2})^{2}}\ &=\frac{54+6t^{2}-12t^{2}}{(9 + t^{2})^{2}}\ &=\frac{54 - 6t^{2}}{(9 + t^{2})^{2}} \end{align*} ]

Answer:

$\frac{54 - 6t^{2}}{(9 + t^{2})^{2}}$