Derivative Rules III

The Quotient Rule

This approach to the quotient rule is credited to Maria Gaetana Agnesi (1718 – 1799) who wrote the first known mathematics textbook Analytical Institutions (1748) to help her brothers learn algebra.

The quotient rule can also be proven from the definition of derivative. But here is a simpler approach – as a corollary of the product rule.

Begin by letting \displaystyle  h\left( x \right)=\frac{f\left( x \right)}{g\left( x \right)}.

Then

f\left( x \right)=h\left( x \right)g\left( x \right) and {f}'\left( x \right)=g\left( x \right){h}'\left( x \right)+h\left( x \right){g}'\left( x \right).

Then solving for {h}'\left( x \right):

\displaystyle {h}'\left( x \right)=\frac{{f}'\left( x \right)-h\left( x \right){g}'\left( x \right)}{g\left( x \right)}

\displaystyle =\frac{{f}'\left( x \right)-\frac{f\left( x \right)}{g\left( x \right)}{g}'\left( x \right)}{g\left( x \right)}

\displaystyle =\frac{g\left( x \right){f}'\left( x \right)-f\left( x \right){g}'\left( x \right)}{{{\left( g\left( x \right) \right)}^{2}}}

Mnemonics

I’m not really one for mnemonics. I cannot spell SOHCOHTOA without saying to myself, “sine, opposite over hypotenuse; cosine, adjacent …” It seems better to me anyway to have student just memorize the formulas in words using the correct terms:

The derivative of a product is the first factor times the derivative of the second plus the second factor times the derivative of the first.

The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all divided by the square of the denominator.

But whatever works for you. Lo Di Hi