find f(t) if f(t)= - 5t^3 + 4t + 9. f(t)=

Explanation:

Step1: Apply power - rule to first term

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For $y=-5t^{3}$, $a=-5$ and $n = 3$. So the derivative of $-5t^{3}$ is $-5\times3t^{3 - 1}=-15t^{2}$.

Step2: Apply power - rule to second term

For $y = 4t$, $a = 4$ and $n = 1$. Using the power - rule, the derivative of $4t$ is $4\times1t^{1 - 1}=4$.

Step3: Derivative of constant

The derivative of a constant $y = c$ (where $c$ is a constant) is 0. For $y = 9$, the derivative is 0.

Step4: Combine derivatives

$f^\prime(t)$ is the sum of the derivatives of each term. So $f^\prime(t)=-15t^{2}+4 + 0=-15t^{2}+4$.

Answer:

$-15t^{2}+4$