When No Need to Apply Quotient Rule for Differentiating a Fraction

The following examples show that you do not need to apply the quotient rule for differentiating when the denominator is constant.

Please see the following cases with the same question.

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$\textit{Application of the quotient rule}$

\( \begin{align} \displaystyle
\dfrac{d}{dx}\dfrac{48x-4x^3}{3} &= \frac{{\frac{d}{{dx}}\left( {48x-4{x^3}} \right) \times 3-\left( {48x-4{x^3}} \right) \times \frac{d}{{dx}}3}}{{{3^2}}} \\
&= \frac{{\left( {48-12{x^2}} \right) \times 3-\left( {48x-4{x^3}} \right) \times 0}}{9} \\
&= \frac{{\left( {48-12{x^2}} \right) \times 3-0}}{9} \\
&= \frac{{\left( {48-12{x^2}} \right) \times 3}}{9} \\
&= \frac{{48-12{x^2}}}{3}
\end{align} \)

$\textit{Just using the simple differentiation rule, then;}$

\( \begin{align} \displaystyle
\dfrac{d}{dx}\dfrac{48x-4x^3}{3} &= \dfrac{1}{3}\dfrac{d}{dx}(48x-4x^3) \\
&= \dfrac{1}{3}(48-12x^2) \\
&= \frac{{48-12{x^2}}}{3}
\end{align} \)

They produce the same results, so there is no need to apply the quotient rule in case the denominator is constant. You notice that applying the quotient rule takes longer than the simple rule, which may result in you being exposed to make careless or silly mistakes as well.


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