Derivative of a polynomial using linearity rules

Apply the sum/difference rule to differentiate each term separately: $g'(t) = \frac{d}{dt}[-6t^5] + \frac{d}{dt}[4t^3] - \frac{d}{dt}[8t] + \frac{d}{dt}[10]$. Using the power rule $\frac{d}{dt}[t^n] = n t^{n-1}$ along with the constant multiple rule, we get $\frac{d}{dt}[-6t^5] = -30t^4$, $\frac{d}{dt}[4t^3] = 12t^2$, and $\frac{d}{dt}[-8t] = -8$.

The derivative of the constant term $10$ is $0$ by the constant rule. Combining all the pieces gives $g'(t) = -30t^4 + 12t^2 - 8$.