Quotient Rule for Derivatives
The Quotient Rule says that the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator,
\bigg( \dfrac{f(x)}{g(x)}\bigg) ' =\dfrac{d}{dx}\bigg( \dfrac{f(x)}{g(x)} \bigg) = \lim\limits_{\Delta x \to0} \dfrac{ \dfrac{f(x+\Delta x)}{g(x+\Delta x)}- \dfrac{f(x)}{g(x)} }{\Delta x}
= \lim\limits_{\Delta x \to0} \dfrac{f(x+\Delta x)g(x)-g(x+\Delta x)f(x)}{\Delta x g(x)g(x+\Delta x)}
= \lim\limits_{\Delta x \to0} \dfrac{f(x+\Delta x)g(x)-g(x+\Delta x)f(x)}{\Delta x} \dfrac{1}{g(x)g(x+\Delta x)}
= \lim\limits_{\Delta x \to0} \dfrac{f(x+\Delta x)g(x)-f(x)g(x)+f(x)g(x)-g(x+\Delta x)f(x)}{\Delta x} \lim\limits_{\Delta x \to0}\dfrac{1}{g(x)g(x+\Delta x)}
= \bigg[ \lim\limits_{\Delta x \to0} \dfrac{f(x+\Delta x )-f(x)}{\Delta x } g(x) - \lim\limits_{\Delta x \to0} \dfrac{g(x+\Delta x )-g(x)}{\Delta x } f(x)\bigg] \dfrac{1}{g^2(x)}
=\dfrac{f'(x)g(x) - g'(x)f(x)}{g^2(x)}
or