find f(t) if f(t)= - 4t^3 + 8t + 1. f(t)=

Explanation:

Step1: Apply power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For $y=-4t^{3}$, $a=-4$ and $n = 3$. So the derivative of $-4t^{3}$ is $-4\times3t^{3 - 1}=-12t^{2}$.

Step2: Differentiate the linear term

For $y = 8t$, using the power - rule with $a = 8$ and $n = 1$, the derivative is $8\times1t^{1 - 1}=8$.

Step3: Differentiate the constant term

The derivative of a constant $y = 1$ (where $a = 1$ and $n = 0$) is $1\times0t^{0 - 1}=0$.

Step4: Combine the derivatives

$f^\prime(t)$ is the sum of the derivatives of each term. So $f^\prime(t)=-12t^{2}+8+0$.

Answer:

$-12t^{2}+8$