# Differentiate: H(u) u sqrt u u sqrt u

## Question:

Differentiate:

{eq}H(u) = (u - \sqrt{u})(u + \sqrt{u}) {/eq}

## Differentiation:

Differentiation is most commonly known as derivative that is used to find the derivative of function with respect to the parameter for a single variable. The product rule is used if we have two different variables.

## Answer and Explanation:

_Given Information

The given expression is:

{eq}H\left( \mu \right) = \left( {\mu - \sqrt \mu } \right)\left( {\mu + \sqrt \mu } \right) {/eq}

By using the identity;

{eq}{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right) {/eq}

Here {eq}a = \mu {/eq} and {eq}b = \sqrt \mu {/eq}

So,

{eq}\begin{align*} H\left( \mu \right) &= \left( {\mu - \sqrt \mu } \right)\left( {\mu + \sqrt \mu } \right)\\ H\left( \mu \right) &= {\left( \mu \right)^2} - {\left( {\sqrt \mu } \right)^2}\\ H\left( \mu \right) &= {\mu ^2} - \mu \end{align*} {/eq}

Differentiating the following expression as follows:

{eq}\begin{align*} \dfrac{d}{{d\mu }}H\left( \mu \right) &= \dfrac{d}{{d\mu }}\left( {{\mu ^2} - \mu } \right)\\ &= \dfrac{d}{{d\mu }}\left( {{\mu ^2}} \right) - \dfrac{d}{{d\mu }}\left( \mu \right)\\ &= 2\mu - 1 \end{align*} {/eq}

The differentiation of function {eq}H\left( \mu \right) \ is \ 2\mu - 1. {/eq}