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