\(\displaystyle
\frac{\mathrm{d}}{\mathrm{d}x}\biggl(kf(x) \biggr) = k \frac{\mathrm{d}}{\mathrm{d}x}f(x)
\)
\(\displaystyle
\frac{\mathrm{d}}{\mathrm{d}x}\biggl(f(x) \pm g(x) \biggr) = \frac{\mathrm{d}}{\mathrm{d}x} f(x) \pm \frac{\mathrm{d}}{\mathrm{d}x} g(x)
\)
\(\displaystyle
\frac{\mathrm{d}}{\mathrm{d}x}\biggl( x^n \biggr) = n x^{n-1}
\)
\(\displaystyle
% \frac{\mathrm{d}}{\mathrm{d}x}\biggl( f(x) g(x) \biggr)
% = \frac{\mathrm{d}}{\mathrm{d}x}\biggl( f(x)\biggr) g(x)
% + f(x) \frac{\mathrm{d}}{\mathrm{d}x}\biggl( g(x) \biggr)
% \qquad\text{ or }\qquad
\Bigl( {\color{DarkGreen} f}{\color{Maroon} g} \Bigr)' = {\color{DarkGreen}f}'{\color{Maroon} g} + {\color{DarkGreen} f}{\color{Maroon} g}'
\)
\(\displaystyle
% \frac{\mathrm{d}}{\mathrm{d}x}\biggl( \tfrac{f(x)}{g(x)} \biggr)
% = \frac{\frac{\mathrm{d}}{\mathrm{d}x}\Bigl( f(x)\Bigr) g(x)
% - f(x) \frac{\mathrm{d}}{\mathrm{d}x}\Bigl( g(x) \Bigr)}{\bigl(g(x)\bigr)^2}
% \qquad\text{ or }\qquad
\Bigl( \tfrac{{\color{DarkGreen} f}}{{\color{Maroon} g}} \Bigr)' = \frac{{\color{DarkGreen}f}'{\color{Maroon} g} - {\color{DarkGreen} f}{\color{Maroon} g}'}{{\color{Maroon} g}^2}
\)
\(\displaystyle
%\frac{\mathrm{d}}{\mathrm{d}x}\biggl( f\bigl({\color{DarkGreen} g(x)}\bigr) \biggr)
%= \frac{\mathrm{d}}{\mathrm{d}x}f\bigl( {\color{DarkGreen} g(x)} \bigr)
%{\color{DarkGreen} \frac{\mathrm{d}}{\mathrm{d}x}g(x) }
%\qquad\text{ or }\qquad
%\Bigl(f\bigl({\color{DarkGreen} g(x)}\bigr)\Bigr)' = f'\bigl({\color{DarkGreen} g(x) }\bigr){\color{DarkGreen} g'(x) }
%\qquad\text{ or }\qquad
\bigl(f \circ {\color{DarkGreen} g}\bigr)' = (f' \circ {\color{DarkGreen} g })({\color{DarkGreen} g' })
\)
\(\displaystyle
\frac{\mathrm{d}}{\mathrm{d}x}\biggl( \mathrm{e}^x \biggr) = \mathrm{e}^x
\)
\(\displaystyle
\frac{\mathrm{d}}{\mathrm{d}x}\biggl( b^x \biggr) = \ln(b)b^x
\)
\(\displaystyle
\frac{\mathrm{d}}{\mathrm{d}x}\biggl( \ln(x) \biggr) = \frac{1}{x}
\)
\(\displaystyle
\frac{\mathrm{d}}{\mathrm{d}x}\biggl( \log_b(x) \biggr) = \frac{1}{\ln(b)x}
\)
\(\displaystyle
\frac{\mathrm{d}}{\mathrm{d}x}\biggl( \sin(x) \biggr) = \cos(x)
\)
\(\displaystyle
\frac{\mathrm{d}}{\mathrm{d}x}\biggl( \cos(x) \biggr) = -\sin(x)
\)