Question

hw11 differentiation rules ii (target c1, c2, c5; §3.3)
score: 1/8 answered: 1/8
question 2
if (f(t)=(t^{2}+4t + 6)(6t^{2}+5)), find (f(t)).
f(t)=
question help: video message instructor
submit question

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Let $u=t^{2}+4t + 6$ and $v = 6t^{2}+5$.

Step2: Find $u'$

Differentiate $u=t^{2}+4t + 6$ with respect to $t$. Using the power rule $\frac{d}{dt}(t^{n})=nt^{n - 1}$, we have $u'=\frac{d}{dt}(t^{2})+\frac{d}{dt}(4t)+\frac{d}{dt}(6)=2t + 4$.

Step3: Find $v'$

Differentiate $v = 6t^{2}+5$ with respect to $t$. Using the power rule, $v'=\frac{d}{dt}(6t^{2})+\frac{d}{dt}(5)=12t$.

Step4: Calculate $f'(t)$

Substitute $u$, $u'$, $v$, and $v'$ into the product - rule formula:

$$
LATEXBLOCK0
$$

Answer:

$24t^{3}+72t^{2}+82t + 20$