Quotient Rule

$f(x)\; =\; \backslash frac\{g(x)\}\{h(x)\}$

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

$=\; \backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; \backslash frac\{1\}\{\backslash Delta\; x\}\; \backslash leftx)h(x)-g(x)h(x+\backslash Delta\; x)\}\{h(x)h(x+\backslash Delta\; x)\}\; \backslash right$

$=\; \backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; \backslash frac\{1\}\{\backslash Delta\; x\}\; \backslash leftx)h(x)-g(x)h(x)-x)-g(x)h(x)\}\{h(x)h(x+\backslash Delta\; x)\}\; \backslash right]$

$=\; \backslash lim\_\{\backslash Delta\; x\; \backslash to\; 0\}\; \backslash frac\{1\}\{\backslash Delta\; x\}\; \backslash leftx)-g(x)-g(x)x)-h(x)\}\{h(x)h(x+\backslash Delta\; x)\}\; \backslash right]$

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

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

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

From the Product Rule

$\backslash mbox\{let\; \}f(x)=\backslash frac\{g(x)\}\{h(x)\}$

$g(x)=f(x)h(x)\backslash mbox\{\; \}$

$g\text{'}(x)=f\text{'}(x)h(x)\; +\; f(x)h\text{'}(x)\backslash mbox\{\; \}$

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

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

Mnemonic

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