Quotient rule
lastmod 2018-08-22created 2015-12-303 edits
In calculus, the quotient rule is a formula for differentiating fractional functions.
If the function \(f(x)\) to be differentiated is as follows:
\[ f(x) = \frac{g(x)}{h(x)} \]
(and if \(h(x)\) ≠ \(0\)), the quotient rule defines the derivative of \(f(x)\) as:
\[ \frac{d}{dx}f(x) = f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{{h(x)}^2} \]
Defined more precisely, for \(x\) in every open set containing \(a\), \(h(a)\) ≠ \(0\) holds and \(h'(a)\) and \(f'(a)\) exist.
\[ f'(a)=\frac{g'(a)h(a) - g(a)h'(a)}{h(a)^2} \]
Proof
It is proved as follows using the product rule.
If \(f(x) = \frac {g(x)} {h(x)}\), then
\[ g(x)=f(x)h(x) \]
\[ g'(x)=f'(x)h(x) + f(x)h'(x) \]
The rest works like solving a simple equation: make \(f'(x)\) the only term on the left side of the equation, and eliminate \(f(x)\) from the right side.
\[ \begin{align} f'(x) &=\frac{g'(x) - f(x)h'(x)}{h(x)} \\ &= \frac{g'(x) - \frac{g(x)}{h(x)}\cdot h'(x)}{h(x)} \\ &= \frac{g'(x)h(x) - g(x)h'(x)}{\left(h(x)\right)^2} \end{align} \]
Example
The derivative of \(\frac{(4x - 2)}{(x^2 + 1)}\) is:
\[ \begin{align} \frac{d}{dx} \frac{(4x - 2)}{x^2 + 1} &= \frac{(x^2 + 1)(4) - (4x - 2)(2x)}{(x^2 + 1)^2} \\ &= \frac{(4x^2 + 4) - (8x^2 - 4x)}{(x^2 + 1)^2} \\ &= \frac{-4x^2 + 4x + 4}{(x^2 + 1)^2} \end{align} \]
In the example above, \(g(x)\) and \(h(x)\) were set as follows:
\[ g(x) = 4x - 2 \]
\[ h(x) = x^2 + 1 \]