Answer
$P'\left( w \right) = 3\sqrt w - \frac{1}{2}{w^{ - 1/2}} - 2{w^{ - 3/2}}$
Work Step by Step
$$\eqalign{
& P\left( w \right) = \frac{{2{w^2} - w + 4}}{{\sqrt w }} \cr
& {\text{Rewrite the function}}{\text{, distribute the numerator}} \cr
& P\left( w \right) = \frac{{2{w^2}}}{{\sqrt w }} - \frac{w}{{\sqrt w }} + \frac{4}{{\sqrt w }} \cr
& P\left( w \right) = \frac{{2{w^2}}}{{{w^{1/2}}}} - \frac{w}{{{w^{1/2}}}} + \frac{4}{{{w^{1/2}}}} \cr
& {\text{Simplify}} \cr
& P\left( w \right) = 2{w^{3/2}} - {w^{1/2}} + 4{w^{ - 1/2}} \cr
& {\text{Differentiating}} \cr
& P'\left( w \right) = \frac{d}{{dw}}\left[ {2{w^{3/2}} - {w^{1/2}} + 4{w^{ - 1/2}}} \right] \cr
& {\text{Use the sum difference rules for derivatives}} \cr
& P'\left( w \right) = \frac{d}{{dw}}\left[ {2{w^{3/2}}} \right] - \frac{d}{{dw}}\left[ {{w^{1/2}}} \right] + \frac{d}{{dw}}\left[ {4{w^{ - 1/2}}} \right] \cr
& P'\left( w \right) = 2\frac{d}{{dw}}\left[ {{w^{3/2}}} \right] - \frac{d}{{dw}}\left[ {{w^{1/2}}} \right] + 4\frac{d}{{dw}}\left[ {{w^{ - 1/2}}} \right] \cr
& {\text{Use the power rule }}\frac{d}{{dw}}\left[ {{w^n}} \right] = n{w^{n - 1}} \cr
& P'\left( w \right) = 2\left( {\frac{3}{2}{w^{3/2 - 1}}} \right) - \frac{1}{2}{w^{1/2 - 1}} + 4\left( { - \frac{1}{2}{w^{ - 1/2 - 1}}} \right) \cr
& P'\left( w \right) = 3{w^{1/2}} - \frac{1}{2}{w^{ - 1/2}} - 2{w^{ - 3/2}} \cr
& {\text{Rewriting}} \cr
& P'\left( w \right) = 3\sqrt w - \frac{1}{2}{w^{ - 1/2}} - 2{w^{ - 3/2}} \cr} $$
