For constants \({\color{Maroon}k}\) and \({\color{Maroon}n},\) and differentiable functions \({\color{SteelBlue}f}\) and \({\color{DarkGoldenRod}g}:\)
\(\displaystyle \quad\qquad\Bigl({\color{SteelBlue}f} \pm {\color{DarkGoldenRod}g} \Bigr)' = {\color{SteelBlue}f}' \pm {\color{DarkGoldenRod}g}' \quad\qquad\)
Differentiation breaks across sums/differences
\(\displaystyle \qquad\Bigl({\color{Maroon}k} {\color{SteelBlue}f} \Bigr)' = {\color{Maroon}k} {\color{SteelBlue}f}'\qquad \)
Constants \({\color{Maroon}k}\) factor out
\(\displaystyle \Bigl( x^{\color{Maroon}n} \Bigr)' = {\color{Maroon}n} x^{\color{Maroon}n-1} \)
The power rule
\(\displaystyle \Bigl( {\color{SteelBlue} f}{\color{DarkGoldenRod} g} \Bigr)' = {\color{SteelBlue}f}'{\color{DarkGoldenRod} g} + {\color{SteelBlue} f}{\color{DarkGoldenRod} g}' \)
The product rule
\(\displaystyle \Bigl( \tfrac{{\color{SteelBlue} f}}{{\color{DarkGoldenRod} g}} \Bigr)' = \frac{{\color{SteelBlue}f}'{\color{DarkGoldenRod} g} - {\color{SteelBlue} f}{\color{DarkGoldenRod} g}'}{{\color{DarkGoldenRod} g}^2} \)
The quotient rule