Question

Differentiate.

\(H\left( u \right) = \left( {u - \sqrt u } \right)\left( {u + \sqrt u } \right)\)

Short answer

Derivative of the function is \(2u - 1\).

Step-by-step solution

Precise Definition of differentiation

The derivative is a function of a real variable. It is the rate of change of output value with respect to an input value. The derivative function is a single variable at a selected input value.

The process of the derivative is called differentiation.

Find the derivative of H using the product rule

The equation forproduct ruleis \({\left( {fg} \right)^\prime } = f'g + gf'\).

Apply this rule on this question as:

\(\begin{aligned}\frac{d}{{du}}H\left( u \right) &= \frac{d}{{du}}\left( {\left( {u - \sqrt u } \right)\left( {u + \sqrt u } \right)} \right)\\ &= \left( {u + \sqrt u } \right)\frac{d}{{du}}\left( {u - \sqrt u } \right) + \left( {u - \sqrt u } \right)\frac{d}{{du}}\left( {u + \sqrt u } \right)\end{aligned}\)

Simplify the obtained condition

Apply derivative rule and find the answer as:

\(\begin{aligned}H'\left( u \right) &= \left( {u + \sqrt u } \right)\frac{d}{{du}}\left( {u - \sqrt u } \right) + \left( {u - \sqrt u } \right)\frac{d}{{du}}\left( {u + \sqrt u } \right)\\ &= \left( {u + \sqrt u } \right)\left( {1 - \frac{1}{{2\sqrt u }}} \right) + \left( {u - \sqrt u } \right)\left( {1 + \frac{1}{{2\sqrt u }}} \right)\\ &= \left( {u + \sqrt u - \frac{1}{2}\sqrt u - \frac{1}{2}} \right) + \left( {u - \sqrt u + \frac{1}{2}\sqrt u - \frac{1}{2}} \right)\\ &= 2u - 1\end{aligned}\)

So, \(H'(u) = 2u - 1\) is the final answer of the derivative function.