Question

find $f(t)$ if $f(t) = 3t^3 - 7t + 2$.
$f(t) = \square$

Explanation:

Step1: Differentiate term by term

Differentiate \(3t^3\), \( - 7t\), and \(2\) separately.
Use the power rule \(\frac{d}{dt}(t^n)=nt^{n - 1}\).
For \(y = 3t^3\), \(\frac{d}{dt}(3t^3)=3\times3t^{3 - 1}=9t^2\).
For \(y=-7t\), \(\frac{d}{dt}(-7t)=-7\times1t^{1 - 1}=-7\).
For \(y = 2\) (a constant), \(\frac{d}{dt}(2)=0\).

Step2: Combine the results

\(f^{\prime}(t)=\frac{d}{dt}(3t^3)-\frac{d}{dt}(7t)+\frac{d}{dt}(2)\)
\(f^{\prime}(t)=9t^2-7 + 0\)

Answer:

\(9t^2-7\)