The Product & Quotient Rules of Differentiation

The derivative of the product of two functions is not the product of their individual derivatives. The product rule of differentiation, also sometimes called the Leibniz rule, says that for two functions \(f\) and \(g,\) \[ \frac{\mathrm{d}}{\mathrm{d}x}\biggl( f(x) g(x) \biggr) = \frac{\mathrm{d}}{\mathrm{d}x}\biggl( f(x)\biggr) g(x) + f(x) \frac{\mathrm{d}}{\mathrm{d}x}\biggl( g(x) \biggr) \qquad\text{ or }\qquad \Bigl( {\color{DarkGreen} f}{\color{Maroon} g} \Bigr)' = {\color{DarkGreen}f}'{\color{Maroon} g} + {\color{DarkGreen} f}{\color{Maroon} g}' \,. \]

Similarly the derivative of the quotient of two functions is not the quotient of their individual derivatives. The quotient rule of differentiation says that for two functions \(f\) and \(g,\) \[ \frac{\mathrm{d}}{\mathrm{d}x}\biggl( \tfrac{f(x)}{g(x)} \biggr) = \frac{\frac{\mathrm{d}}{\mathrm{d}x}\Bigl( f(x)\Bigr) g(x) - f(x) \frac{\mathrm{d}}{\mathrm{d}x}\Bigl( g(x) \Bigr)}{\bigl(g(x)\bigr)^2} \qquad\text{ or }\qquad \Bigl( \tfrac{{\color{DarkGreen} f}}{{\color{Maroon} g}} \Bigr)' = \frac{{\color{DarkGreen}f}'{\color{Maroon} g} - {\color{DarkGreen} f}{\color{Maroon} g}'}{{\color{Maroon} g}^2} \,. \]

Examples

\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \biggl( {\color{DarkGreen} \bigl(x^3-1\bigr) }{\color{Maroon} \sqrt{x+2} } \biggr) = {\color{DarkGreen} \bigl(3x^2\bigr) }{\color{Maroon} \sqrt{x+2} } + \dfrac{{\color{DarkGreen} x^3-1 }}{{\color{Maroon} 2\sqrt{x+2}}} \)

\(\displaystyle \frac{\mathrm{d}}{\mathrm{d}x} \biggl( \frac{{\color{DarkGreen} \bigl(x^3-1\bigr) }}{{\color{Maroon} \sqrt{x+2}} } \biggr) = \frac{{\color{DarkGreen} \bigl(3x^2\bigr) }{\color{Maroon} \sqrt{x+2} } - \dfrac{{\color{DarkGreen} x^3-1 }}{{\color{Maroon} 2\sqrt{x+2}}}}{{\color{Maroon} x+2}} \)