Question

find \\( \frac { d y } { d t } \\).

\\( y = 3 t left( 2 t ^ { 2 } - 7
ight) ^ { 4 } \\)

\\( \frac { d y } { d t } = \\)

Explanation:

Step1: Apply the product rule

The product rule states that if \(y = u\cdot v\), then \(\frac{dy}{dt}=u'\cdot v + u\cdot v'\). Let \(u = 3t\) and \(v=(2t^{2}-7)^{4}\). First, find \(u'\): \(u'=\frac{d}{dt}(3t)=3\).

Step2: Apply the chain rule to find \(v'\)

The chain rule states that if \(y = f(g(t))\), then \(y'=f'(g(t))\cdot g'(t)\). Let \(g(t)=2t^{2}-7\), \(f(g)=g^{4}\). \(g'(t) = 4t\), \(f'(g)=4g^{3}\). So \(v'=\frac{d}{dt}(2t^{2}-7)^{4}=4(2t^{2}-7)^{3}\cdot4t = 16t(2t^{2}-7)^{3}\).

Step3: Substitute \(u\), \(u'\), \(v\), \(v'\) into the product rule formula

\(\frac{dy}{dt}=u'\cdot v+u\cdot v'=3(2t^{2}-7)^{4}+3t\cdot16t(2t^{2}-7)^{3}\).

Step4: Simplify the expression

Factor out \(3(2t^{2}-7)^{3}\):

$$
LATEXBLOCK0
$$

Answer:

\(3(2t^{2}-7)^{3}(18t^{2}-7)\)