Product and Quotient Rules

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Product and Quotient Rules

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The Quotient Rule

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The Quotient Rule

The derivative of a quotient is not the derivative of the numerator divided by the derivative of the denominator. The video below shows this with an example. Instead,

The Quotient Rule

$\frac{d}{dx}\left( \frac{f(x)}{g(x)}\right) = \frac{g(x) f'(x) - f(x) g'(x)}{[g(x)]^2}.$

The quotient rule can be derived from the product rule. If we write $\displaystyle f(x) = g(x)\cdot \frac{f(x)}{g(x)}$ , then the product rule says that

$f'(x) = \left ( g(x) \cdot\frac{f(x)}{g(x)} \right )',$ i.e.,

$f'(x)= g'(x) \frac{f(x)}{g(x)} + g(x) \left ( \frac{f(x)}{g(x)} \right )'.$ Solving for

$\left( \frac{f(x)}{g(x)} \right )'$ gives

$\left ( \frac{f(x)}{g(x)} \right )' = \frac{f'(x) - g'(x)\frac{f(x)}{g(x)}}{g(x)} = \frac{g(x) f'(x) - f(x) g'(x)}{[g(x)]^2}.$

The most common mistake with this rule is mix up the signs of the $g f'$ and $f g'$ terms. If $f(x)$ and $g(x)$ are positive, then increasing the numerator $f(x)$ will increase the ratio $f/g$ , so the $g f'$ term must be positive. However, increasing the denominator $g(x)$ will decrease the ratio, so the $f g'$ term must be negative.