Question

Differentiate.

\(J\left( u \right) = \left( {\frac{1}{u} + \frac{1}{{{u^2}}}} \right)\left( {u + \frac{1}{u}} \right)\)

Short answer

Derivative of the function is \( - \left( {\frac{1}{{{u^2}}} + \frac{2}{{{u^3}}} + \frac{3}{{{u^4}}}} \right)\)\(\).

Step-by-step solution

Precise Definition of differentiation

A 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.

Find the derivative of J using the Product rule

The equation for the Product rule is \({\left( {fg} \right)^\prime } = fg' + gf'\).

\(\begin{aligned}\frac{d}{{du}}\left( {J\left( u \right)} \right) &= \frac{d}{{du}}\left( {\left( {\frac{1}{u} + \frac{1}{{{u^2}}}} \right)\left( {u + \frac{1}{u}} \right)} \right)\\ &= \left( {\frac{1}{u} + \frac{1}{{{u^2}}}} \right)\frac{d}{{du}}\left( {u + \frac{1}{u}} \right) + \left( {u + \frac{1}{u}} \right)\frac{d}{{du}}\left( {\frac{1}{u} + \frac{1}{{{u^2}}}} \right)\end{aligned}\)

Simplify the obtained condition

Simplify the obtained derivative as:

\(\begin{aligned}J'\left( u \right) &= \left( {\frac{1}{u} + \frac{1}{{{u^2}}}} \right)\frac{d}{{du}}\left( {u + \frac{1}{u}} \right) + \left( {u + \frac{1}{u}} \right)\frac{d}{{du}}\left( {\frac{1}{u} + \frac{1}{{{u^2}}}} \right)\\ &= \left( {\frac{1}{u} + \frac{1}{{{u^2}}}} \right)\left( {1 - \frac{1}{{{u^2}}}} \right) + \left( {u + \frac{1}{u}} \right)\left( { - \frac{1}{{{u^2}}} - \frac{2}{{{u^3}}}} \right)\\ &= \left( {\frac{1}{u} + \frac{1}{{{u^2}}} - \frac{1}{{{u^3}}} - \frac{1}{{{u^4}}}} \right) + \left( { - \frac{1}{u} - \frac{1}{{{u^3}}} - \frac{2}{{{u^2}}} - \frac{2}{{{u^4}}}} \right)\\ &= \frac{1}{u} + \frac{1}{{{u^2}}} - \frac{1}{{{u^3}}} - \frac{1}{{{u^4}}} - \frac{1}{u} - \frac{1}{{{u^3}}} - \frac{2}{{{u^2}}} - \frac{2}{{{u^4}}}\\ &= - \frac{1}{{{u^2}}} - \frac{2}{{{u^3}}} - \frac{3}{{{u^4}}}\\ &= - \left( {\frac{1}{{{u^2}}} + \frac{2}{{{u^3}}} + \frac{3}{{{u^4}}}} \right)\end{aligned}\)

So, \(J'(u) = - \left( {\frac{1}{{{u^2}}} + \frac{2}{{{u^3}}} + \frac{3}{{{u^4}}}} \right)\)is the final answer of this derivative.