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The Quotient Rule

Now, we want to be able to take the derivative of a fraction like f/g, where f and g are two functions. This one is a little trickier to remember, but luckily it comes with its own song. The formula is as follows:

\begin{displaymath}\boxed{ \frac {d} {dx} (\frac{f}{g}) = \frac {f'g - f g'} {g^2} }\end{displaymath}

How to Remember this Formula (with thanks to Snow White  and the Seven Dwarves):


Replacing f by hi and g by ho (hi for high up there in the numerator and ho for low down there in the denominator), and letting D stand in for `the derivative of', the formula becomes:


\begin{displaymath}D\frac{hi}{ho} = \frac { hoD(hi)- hiD(ho)} { ho^2 }
\end{displaymath}

In words, that is ``ho dee hi minus hi dee ho over ho ho". Now, if Sleepy and Sneezy can remember that, it shouldn't be any problem for you.

As an example,


\begin{displaymath}\frac {d} {dx} (\frac{x}{x^2+1}) = \frac{ (x^2+1)(1) - x(2x) }{(x^2+1)^2}
= \frac{1-x^2}{(x^2+1)^2} .
\end{displaymath}

A Common Mistake: Remembering the quotient rule  wrong and getting an extra minus sign in the answer. It's very easy to forget whether it's ho dee hi first (yes, it is) or hi dee ho first (no, it's not).



Next: Derivatives of Trig Functions Up: Derivatives: How to Find Previous: The Product Rule
Joel Hass
1999-05-26