Question

3-34:Differentiate the function.

33. \(P\left( w \right) = \frac{{2{w^2} - w + 4}}{{\sqrt w }}\)

Short answer

The derivative of the given function is \(\frac{{dP}}{{dw}} = 3\sqrt w - \frac{1}{{2\sqrt w }} - \frac{2}{{w\sqrt w }}\).

Step-by-step solution

Different rules of differentiation

The given function is \(P\left( w \right) = \frac{{2{w^2} - w + 4}}{{\sqrt w }}\).

The sum rule is:\(\frac{d}{{dx}}\left( {g\left( x \right) + f\left( x \right)} \right) = \frac{d}{{dx}}\left( {g\left( x \right)} \right) + \frac{d}{{dx}}\left( {f\left( x \right)} \right)\)

And, the power rule is: \(\frac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\)

Evaluating Derivative using all these rules

Differentiating with respect to\(w\)as:

\(\begin{align}P'\left( w \right) &= \frac{d}{{dw}}\left( {\frac{{2{w^2} - w + 4}}{{\sqrt w }}} \right)\\ &= \frac{d}{{dw}}\left( {2w\sqrt w - \sqrt w - \frac{4}{{\sqrt w }}} \right)\\ &= 2\left( {\frac{3}{2}\sqrt w } \right) - \frac{1}{{2\sqrt w }} + \frac{2}{{w\sqrt w }}\\ &= 3\sqrt w - \frac{1}{{2\sqrt w }} - \frac{2}{{w\sqrt w }}\end{align}\)

Hence, the derivative of the given function is \(\frac{{dP}}{{dw}} = 3\sqrt w - \frac{1}{{2\sqrt w }} - \frac{2}{{w\sqrt w }}\).