differentiate. f(t) = 5t / (5 + t^2) f(t) =

Explanation:

Step1: Identify quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = 5t$, $v=5 + t^{2}$.

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

Differentiate $u = 5t$ with respect to $t$, $u'=5$. Differentiate $v = 5 + t^{2}$ with respect to $t$, $v' = 2t$.

Step3: Apply quotient - rule

Substitute $u$, $u'$, $v$, and $v'$ into the quotient - rule formula: [ \begin{align*} f'(t)&=\frac{u'v - uv'}{v^{2}}\ &=\frac{5(5 + t^{2})-5t(2t)}{(5 + t^{2})^{2}}\ &=\frac{25+5t^{2}-10t^{2}}{(5 + t^{2})^{2}}\ &=\frac{25 - 5t^{2}}{(5 + t^{2})^{2}} \end{align*} ]

Answer:

$\frac{25 - 5t^{2}}{(5 + t^{2})^{2}}$