Answer
$g'\left( x \right) = 4$
Work Step by Step
$$\eqalign{
& g\left( x \right) = 4x + 7 \cr
& {\text{Differentiate the function}} \cr
& g'\left( x \right) = \frac{d}{{dx}}\left[ {4x + 7} \right] \cr
& \cr
& {\text{Use the sum rule for differentiation }} \cr
& \frac{d}{{dx}}\left[ {f\left( x \right) + g\left( x \right)} \right] = \frac{d}{{dx}}f\left( x \right) + \frac{d}{{dx}}g\left( x \right) \cr
& {\text{then}} \cr
& g'\left( x \right) = \frac{d}{{dx}}\left[ {4x} \right] + \frac{d}{{dx}}\left[ 7 \right] \cr
& \cr
& {\text{Use the constant multiple rule}} \cr
& g'\left( x \right) = 4\frac{d}{{dx}}\left[ x \right] + \frac{d}{{dx}}\left[ 7 \right] \cr
& \cr
& {\text{Compute the derivatives}}{\text{, }}\frac{d}{{dx}}\left[ x \right] = 1{\text{ and }}\frac{d}{{dx}}\left[ c \right] = 0,{\text{ so}} \cr
& g'\left( x \right) = 4\left( 1 \right) + 0 \cr
& {\text{Simplify}} \cr
& g'\left( x \right) = 4 \cr} $$
