5. -/1 points details my notes differentiate. f(t) = 4t / (8 + t^2) f(t) = need help? read it submit answer

Explanation:

Step1: Identify quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = 4t$, $v=8 + t^{2}$.

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

Differentiate $u = 4t$ with respect to $t$, $u'=\frac{d(4t)}{dt}=4$. Differentiate $v = 8 + t^{2}$ with respect to $t$, $v'=\frac{d(8 + t^{2})}{dt}=2t$.

Step3: Apply quotient - rule

$f'(t)=\frac{u'v - uv'}{v^{2}}=\frac{4(8 + t^{2})-4t(2t)}{(8 + t^{2})^{2}}$.

Step4: Simplify the expression

Expand the numerator: $4(8 + t^{2})-4t(2t)=32+4t^{2}-8t^{2}=32 - 4t^{2}$. So $f'(t)=\frac{32 - 4t^{2}}{(8 + t^{2})^{2}}$.

Answer:

$\frac{32 - 4t^{2}}{(8 + t^{2})^{2}}$