DIFFERENTIATION 07
The Quotient Rule
The quotient rule differentiates one changing function divided by another.
The main idea
Suppose
\[ y=\frac{f(x)}{g(x)}, \qquad g(x)\neq0. \]Then
This is the quotient rule.
1. Why a special rule is needed
We cannot generally differentiate the numerator and denominator separately:
\[ \frac{d}{dx} \left( \frac{f}{g} \right) \neq \frac{f'}{g'}. \]A quotient depends on how both the numerator and denominator change.
2. Deriving it from the product rule
Start with
\[ y=\frac{f}{g}. \]Multiply both sides by \(g\):
\[ yg=f. \]Differentiate using the product rule:
\[ y'g+yg'=f'. \]Solve for \(y'\):
\[ y'g=f'-yg'. \]Therefore,
\[ y' = \frac{f'-yg'}{g}. \]Since
\[ y=\frac{f}{g}, \]substitute this into the expression:
\[ y' = \frac{ f'-\frac{f}{g}g' }{g}. \]Simplifying gives
3. A simple way to remember it
For
\[ \frac{f}{g}, \]the numerator of the derivative is
\[ f'g-fg'. \]So:
and then divide by
\[ g^2. \]4. Example
Consider
\[ y=\frac{x^2}{x+1}. \]Let
\[ f(x)=x^2, \qquad g(x)=x+1. \]Then
\[ f'(x)=2x, \qquad g'(x)=1. \]Using the quotient rule,
\[ \frac{dy}{dx} = \frac{ (2x)(x+1)-x^2(1) }{ (x+1)^2 }. \]Simplifying,
\[ \frac{dy}{dx} = \frac{ 2x^2+2x-x^2 }{ (x+1)^2 }. \]Therefore,
5. Example with trigonometric functions
Consider
\[ y=\frac{\sin x}{x}. \]Let
\[ f(x)=\sin x, \qquad g(x)=x. \]Then
\[ f'(x)=\cos x, \qquad g'(x)=1. \]So
\[ \frac{dy}{dx} = \frac{x\cos x-\sin x}{x^2}. \]6. Example with an exponential function
Suppose
\[ y=\frac{e^x}{x}. \]Then
\[ f(x)=e^x, \qquad g(x)=x, \]so
\[ f'(x)=e^x, \qquad g'(x)=1. \]Therefore,
\[ \frac{dy}{dx} = \frac{xe^x-e^x}{x^2}. \]Factorising,
\[ \boxed{ \frac{dy}{dx} = \frac{e^x(x-1)}{x^2} } \]7. Sometimes rewriting is easier
Consider
\[ y=\frac{1}{x^2}. \]We could use the quotient rule, but it is simpler to write
\[ y=x^{-2}. \]Then the power rule gives
\[ \frac{dy}{dx} = -2x^{-3} = -\frac{2}{x^3}. \]So the quotient rule is not always the shortest method.
8. Domain restrictions remain important
If
\[ y=\frac{f(x)}{g(x)}, \]the original function is defined only where
\[ g(x)\neq0. \]Differentiating does not remove this restriction.
For example,
\[ y=\frac{x^2}{x-3} \]is not defined at
\[ x=3. \]9. Quotient rule in Leibniz notation
If
\[ y=\frac{u}{v}, \]where \(u\) and \(v\) depend on \(x\), then
This is the same rule written using differential notation.
10. A common mistake
Do not write
\[ \left(\frac{f}{g}\right)' = \frac{f'}{g'}. \]For example,
\[ \frac{x^2}{x}=x \]for \(x\neq0\).
The derivative is therefore
\[ 1. \]But differentiating numerator and denominator separately would give
\[ \frac{2x}{1}=2x, \]which is wrong.
11. The structure to remember
The order in the numerator matters because subtraction is involved.