differentiate the following function. f(t) = 1/2t^6 - 2t^4 + 3t f(t) =

Explanation:

Step1: Apply power - rule for each term

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For the first term $\frac{1}{2}t^6$, $a=\frac{1}{2}$ and $n = 6$. Its derivative is $\frac{1}{2}\times6t^{6 - 1}=3t^5$. For the second term $-2t^4$, $a=-2$ and $n = 4$, its derivative is $-2\times4t^{4 - 1}=-8t^3$. For the third term $3t$, $a = 3$ and $n = 1$, its derivative is $3\times1t^{1 - 1}=3$.

Step2: Combine the derivatives of each term

$f^\prime(t)$ is the sum of the derivatives of each term in $f(t)$. So $f^\prime(t)=3t^5-8t^3 + 3$.

Answer:

$3t^5-8t^3 + 3$