Question

Differentiate.

\(G\left( u \right) = \frac{{6{u^4} - 5u}}{{u + 1}}\)

Short answer

Derivative of the function is \(\frac{{18{u^4} + 24{u^3} - 5}}{{{{\left( {u + 1} \right)}^2}}}\).

Step-by-step solution

Precise Definition of differentiation

Derivative of a function used in calculus mathematics. It is the change of output value with respect to an input value. The process of finding derivatives is called differentiation.

Find the derivative of G using the Quotient rule

The equation for the Quotient ruleis \({\left( {\frac{f}{g}} \right)^\prime } = \frac{{gf' - fg'}}{{{{\left( g \right)}^2}}}\)

Apply Quotient rule for the function \(G(u) = \frac{{6{u^4} - 5u}}{{u + 1}}\).

\(\frac{d}{{du}}\left( {G\left( u \right)} \right) = \frac{d}{{du}}\left( {\frac{{6{u^4} - 5u}}{{u + 1}}} \right)\)

Simplify the obtained condition

The derivative of \(G\left( u \right)\) can be obtained as:

\(\begin{aligned}G'(u) &= \frac{{\left( {u + 1} \right)\frac{d}{{du}}\left( {6{u^4} - 5u} \right) - \left( {6{u^4} - 5u} \right)\frac{d}{{du}}\left( {u + 1} \right)}}{{{{\left( {u + 1} \right)}^2}}}\\ &= \frac{{\left( {u + 1} \right)\left( {24{u^3} - 5} \right) - \left( {6{u^4} - 5u} \right)\left( 1 \right)}}{{{{\left( {u + 1} \right)}^2}}}\\ &= \frac{{24{u^4} + 24{u^3} - 5u - 5 - \left( {6{u^4} - 5u} \right)}}{{{{\left( {u + 1} \right)}^2}}}\\ &= \frac{{18{u^4} + 24{u^3} - 5}}{{{{\left( {u + 1} \right)}^2}}}\end{aligned}\)

So, \(G'(u) = \frac{{18{u^4} + 24{u^3} - 5}}{{{{\left( {u + 1} \right)}^2}}}\)is the final answer of this derivative function.