Answer
$g'\left( t \right) = 5 + 8t$
Work Step by Step
$$\eqalign{
& g\left( t \right) = 5t + 4{t^2} \cr
& {\text{Differentiate the function}} \cr
& g'\left( t \right) = \frac{d}{{dt}}\left[ {5t + 4{t^2}} \right] \cr
& \cr
& {\text{Use the sum rule for differentiation }} \cr
& \frac{d}{{dt}}\left[ {f\left( t \right) + g\left( t \right)} \right] = \frac{d}{{dt}}f\left( t \right) + \frac{d}{{dt}}g\left( t \right) \cr
& {\text{then}} \cr
& g'\left( t \right) = \frac{d}{{dt}}\left[ {5t} \right] + \frac{d}{{dt}}\left[ {4{t^2}} \right] \cr
& \cr
& {\text{Use the constant multiple rule}} \cr
& g'\left( t \right) = 5\frac{d}{{dt}}\left[ t \right] + 4\frac{d}{{dt}}\left[ {{t^2}} \right] \cr
& \cr
& {\text{Apply the rules: }}\frac{d}{{dt}}\left[ {{t^n}} \right] = n{t^{n - 1}}{\text{ and }}\frac{d}{{dt}}\left[ t \right] = 1 \cr
& g'\left( t \right) = 5\left( 1 \right) + 4\left( {2t} \right) \cr
& {\text{Simplify}} \cr
& g'\left( t \right) = 5 + 8t \cr} $$
