\(\displaystyle {\frac{\mathrm{d}}{\mathrm{d}x} \arcsin(x) = \frac{1}{\sqrt{1-x^2}}}\)
\(\displaystyle {\frac{\mathrm{d}}{\mathrm{d}x} \arccos(x) = -\frac{1}{\sqrt{1-x^2}}}\)
\(\displaystyle {\frac{\mathrm{d}}{\mathrm{d}x} \mathrm{arcsec}(x) = \frac{1}{x\sqrt{x^2-1}}}\)
\(\displaystyle {\frac{\mathrm{d}}{\mathrm{d}x} \mathrm{arccsc}(x) = -\frac{1}{x\sqrt{x^2-1}}}\)
\(\displaystyle {\frac{\mathrm{d}}{\mathrm{d}x} \arctan(x) = \frac{1}{1+x^2}}\)
\(\displaystyle {\frac{\mathrm{d}}{\mathrm{d}x} \mathrm{arccot}(x) = -\frac{1}{1+x^2}}\)
\(\displaystyle {\int \tan(x) \,\mathrm{d}x = \ln|\sec(x)| +C \qquad \int \ln(x) \,\mathrm{d}x = x\ln(x) - x +C} \)
\(\displaystyle {\int \sec(x) \,\mathrm{d}x = \ln|\sec(x) + \tan(x)| +C} \)
\(\displaystyle {\int \sec^3(x) \,\mathrm{d}x = \frac{1}{2}\sec(x)\tan(x) + \frac{1}{2}\int\sec(x)\,\mathrm{d}x} \)
\(\displaystyle {\mathrm{e}^x = \sum_{n=0}^\infty \frac{1}{n!}x^n} \)
\(\displaystyle {\frac{1}{1-x} = \sum_{n=0}^\infty x^n \;\text{ for } |x| \lt 1} \)
\(\displaystyle {\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1}} \)
\(\displaystyle {\cos(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n}} \)
\(\displaystyle {\ln(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}x^n \;\text{ for } |x| \lt 1 } \)
\(\displaystyle {\arctan(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)}x^{2n+1} \;\text{ for } |x| \lt 1} \)