Quotient Rule Derivative

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The Quotient Rule in Calculus

Definition

The quotient rule is a method in calculus used to find the derivative of a function expressed as the ratio of two differentiable functions.

Formula

Let $f(x)$ and $g(x)$ be two differentiable functions. The derivative of their quotient, $h(x)\; =\; \backslash frac\{f(x)\}\{g(x)\}$, is given by:

$h\text{'}(x)\; =\; \backslash frac\{g(x)f\text{'}(x)\; -\; f(x)g\text{'}(x)\}\{g(x)^2\}$

Derivation

The quotient rule can be derived using the definition of the derivative:

$h\text{'}(x)\; =\; \backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; \backslash frac\{h(x\; +\; \backslash Delta\; x)\; -\; h(x)\}\{\backslash Delta\; x\}$

Substituting the definition of $h(x)$, we get:

$h\text{'}(x)\; =\; \backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; \backslash frac\{\backslash frac\{f(x\; +\; \backslash Delta\; x)\}\{g(x\; +\; \backslash Delta\; x)\}\; -\; \backslash frac\{f(x)\}\{g(x)\}\}\{\backslash Delta\; x\}$

Simplifying the expression and applying l'Hôpital's rule, we arrive at the formula for the quotient rule.

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