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.
$\textit{Application of the quotient rule}$
\( \begin{align} \displaystyle
\dfrac{d}{dx}\dfrac{48x-4x^3}{3} &= \frac{{\frac{d}{{dx}}\left( {48x-4{x^3}} \right) \times 3-\left( {48x-4{x^3}} \right) \times \frac{d}{{dx}}3}}{{{3^2}}} \\
&= \frac{{\left( {48-12{x^2}} \right) \times 3-\left( {48x-4{x^3}} \right) \times 0}}{9} \\
&= \frac{{\left( {48-12{x^2}} \right) \times 3-0}}{9} \\
&= \frac{{\left( {48-12{x^2}} \right) \times 3}}{9} \\
&= 16-4x^2
\end{align} \)
$\textit{Just using the simple differentiation rule, then;}$
\( \begin{align} \displaystyle
\dfrac{d}{dx}\dfrac{48x-4x^3}{3} &= \dfrac{1}{3}\dfrac{d}{dx}(48x-4x^3) \\
&= \dfrac{1}{3}(48-12x^2) \\
&= 16-4x^2
\end{align} \)
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.
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume
Responses