Question

evaluate and simplify y. y = 3t^3 sin t y = □

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Here, $u = 3t^{3}$ and $v=\sin t$.

Step2: Differentiate $u$

Differentiate $u = 3t^{3}$ with respect to $t$. Using the power - rule $\frac{d}{dt}(at^{n})=nat^{n - 1}$, we have $u'=\frac{d}{dt}(3t^{3})=9t^{2}$.

Step3: Differentiate $v$

Differentiate $v=\sin t$ with respect to $t$. We know that $\frac{d}{dt}(\sin t)=\cos t$.

Step4: Calculate $y'$

Substitute $u$, $u'$, $v$, and $v'$ into the product - rule formula: $y'=u'v+uv'=9t^{2}\sin t+3t^{3}\cos t$.

Answer:

$9t^{2}\sin t + 3t^{3}\cos t$