Answer
$W'\left( t \right) = \frac{1}{{2\sqrt t }} - 2{e^t}$
Work Step by Step
$$\eqalign{
& W\left( t \right) = \sqrt t - 2{e^t} \cr
& {\text{Rewrite the function}} \cr
& W\left( t \right) = {t^{1/2}} - 2{e^t} \cr
& {\text{Differentiate the function}} \cr
& W'\left( t \right) = \frac{d}{{dt}}\left[ {{t^{1/2}} - 2{e^t}} \right] \cr
& {\text{Use the sum diffence rules for differentiation }} \cr
& W'\left( t \right) = \frac{d}{{dt}}\left[ {{t^{1/2}}} \right] - \frac{d}{{dt}}\left[ {2{e^t}} \right] \cr
& {\text{Use the constant multiple rule}} \cr
& W'\left( t \right) = \frac{d}{{dt}}\left[ {{t^{1/2}}} \right] - 2\frac{d}{{dt}}\left[ {{e^t}} \right] \cr
& {\text{Apply the power rule: }}\frac{d}{{dt}}\left[ {{t^n}} \right] = n{t^{n - 1}}{\text{ and }}\frac{d}{{dt}}\left[ {{e^t}} \right] = {e^t} \cr
& W'\left( t \right) = \frac{1}{2}{t^{1/2 - 1}} - 2\left( {{e^t}} \right) \cr
& W'\left( t \right) = \frac{1}{2}{t^{ - 1/2}} - 2{e^t} \cr
& W'\left( t \right) = \frac{1}{{2\sqrt t }} - 2{e^t} \cr} $$
