Calc2.38

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Derive the Quotient Rule formula.

Recall the Product Rule: {\frac  {d}{dx}}(ab)=ab^{\prime }+a^{\prime }b\,

The Product Rule can be applied to a quotient by noting that {\frac  {a}{b}}=ab^{{-1}}\,.

Also note that the Power Rule and Chain Rule must be applied when taking (b^{{-1}})^{\prime }\,. Therefore,

{\frac  {d}{dx}}\left({\frac  {a}{b}}\right)\, ={\frac  {d}{dx}}\left(ab^{{-1}}\right)\,
=a(b^{{-1}})^{\prime }+a^{\prime }b^{{-1}}\,
=a(-b^{{-2}}b^{\prime })+a^{\prime }b^{{-1}}\,
={\frac  {-ab^{\prime }}{b^{2}}}+{\frac  {a^{\prime }}{b}}\,
={\frac  {-ab^{\prime }}{b^{2}}}+{\frac  {a^{\prime }}{b}}\cdot {\frac  {b}{b}}\,
={\frac  {-ab^{\prime }}{b^{2}}}+{\frac  {ba^{\prime }}{b^{2}}}\,
={\frac  {ba^{\prime }-ab^{\prime }}{b^{2}}}\,

which is the correct Quotient Rule formula.

In practice, it is usually easier to memorize just the Product Rule and apply it in both cases as here, rather than memorize two formulas.

Main Page : Calculus : Quotient Rule