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 a constant.
Please see the following cases with the same question.

1. $\textit{Application of the quotient rule}$
\( \begin{align} \displaystyle
\dfrac{d}{dx}\dfrac{48x – 4x^3}{3} &= \frac{{\frac{d}{{dx}}\left( {48x – 4{x^3}} \right) \times 3 – \left( {48x – 4{x^3}} \right) \times \frac{d}{{dx}}3}}{{{3^2}}} \\
&= \frac{{\left( {48 – 12{x^2}} \right) \times 3 – \left( {48x – 4{x^3}} \right) \times 0}}{9} \\
&= \frac{{\left( {48 – 12{x^2}} \right) \times 3 – 0}}{9} \\
&= \frac{{\left( {48 – 12{x^2}} \right) \times 3}}{9} \\
&= \frac{{48 – 12{x^2}}}{3} \\
\end{align} \)

2. $\textit{Just using the simple differentiation rule, then;}$
\( \begin{align} \displaystyle
\dfrac{d}{dx}\dfrac{48x – 4x^3}{3} &= \dfrac{1}{3}\dfrac{d}{dx}(48x – 4x^3) \\
&= \dfrac{1}{3}(48 – 12x^2) \\
&= \frac{{48 – 12{x^2}}}{3} \\
\end{align} \)

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

Hope this helps!

Note that this is a respond to one of our YouTube students: Maxima and Minima Problems