Skip to main content

Calculus

Section 3.4 The Quotient Rule

Subsection 3.4.1 The Quotient Rule

Subsection 3.4.2 Using the Quotient Rule

The rule for differentiating polynomials, combined with the quotient rule, provides a method for differentiating any rational function (a quotient of polynomials).
Often, the quotient rule will result in fractions with numerators (and sometimes denominators) that can be simplified. This is often helpful, especially in applications, where we'd like to analyze derivatives.
Also, just like the quotient rule, sometimes it is easier to simplify a quotient first before trying to apply the quotient rule. This is particularly true if the denominator is a monomial (one term).

Subsection 3.4.3 Remembering the Quotient Rule

This rule is more difficult to remember, so there are multiple techniques and mnemonics to help remember it. First, notice that the numerator of the quotient rule resembles the product rule, except with a negative sign between the terms. Also,
  • Instead of using \(f\) and \(g\text{,}\) we can use “numerator” and “denominator”.
    \begin{equation*} (\text{quotient})' = \frac{(\text{denominator}) \times (\text{numerator})' - (\text{numerator}) \times (\text{denominator})'}{(\text{denominator})^2} \end{equation*}
    Alternatively, in short form,
    \begin{equation*} (\text{quotient})' = \frac{(\text{denom}) \times (\text{num})' - (\text{num}) \times (\text{denom})'}{(\text{denom})^2} \end{equation*}
  • Omitting the arguments, we have,
    \begin{equation*} \boxed{\brac{\frac{f}{g}}' = \frac{gf' - fg'}{g^2}} \end{equation*}
  • If we consider \(f\) to be the “high” function, \(g\) to be the “low” function, and “di” (pronounced “dee”) representing the derivative, we get,
    \begin{equation*} \boxed{(\text{quotient})' = \frac{\text{low di-high} - \text{high di-low}}{(\text{low})^2}} \end{equation*}
    which is read as “low di-high minus high di-low over low squared”.
  • For a Santa-themed mnemonics, if we take \(f\) to be “hi” (the “high” function) and \(g\) to be the “ho” function (hi-ho), then we get,
    \begin{equation*} (\text{quotient})' = \frac{\text{ho di-hi} - \text{hi di-ho}}{(\text{ho})^2} \end{equation*}
    read as “ho di-hi minus hi di-ho over ho ho”.
<Prev^TopNext>
Authored in PreTeXt
Powered by Runestone