Differentiate x squared
Differentiate x cubed
Differentiate x^n
Differentiate sin x
![{\displaystyle \lim _{h\to 0}{\frac {1-\cos h}{h}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0b89588a53dce0684666db733a7e343f396971)
Multiple Angle
![{\displaystyle \cos 3\theta =?}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2bbcc85f581c352598ef8897f6bb10c9442a0c)
![{\displaystyle \sin 4\theta =?}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ad7dd13cc75ec4d2db0215723a0f3424101dc3e)
![{\displaystyle \cos 12\theta =?}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5dd909ea15c98ea4189004d8a8c866767564ad)
![{\displaystyle e^{ix}=\cos x+i\sin x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4907c0489ab08ce550c7700a1587d4634801dff8)
![{\displaystyle e^{3i\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb1c4525a54b6c98d49680296a4dba65e1ec5cd)
![{\displaystyle e^{3i\theta }={(e^{i\theta })}^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df5d1ef6f4f16775e012e98de7e3c4e1f6f93b57)
![{\displaystyle {\begin{alignedat}{4}\cos 3\theta +i\sin 3\theta \,\,&=\\\cos 3\theta +i\sin 3\theta \,\,&=(\cos \theta +i\sin \theta )^{3}\\&=\cos ^{3}\theta \\&=\cos ^{3}\theta +3\cos ^{2}\theta \,i\sin \theta \\&=\cos ^{3}\theta +3\cos ^{2}\theta \,i\sin \theta +3\cos \theta \,i^{2}\sin ^{2}\theta \\&=\cos ^{3}\theta +3\cos ^{2}\theta \,i\sin \theta +3\cos \theta \,i^{2}\sin ^{2}\theta +i^{3}\sin ^{3}\theta \\&=\cos ^{3}\theta +3\cos ^{2}\theta \,{\color {red}i}\sin \theta +3\cos \theta \,i^{2}\sin ^{2}\theta +i^{3}\sin ^{3}\theta \\&=\cos ^{3}\theta +3\cos ^{2}\theta \,i\sin \theta +3\cos \theta \,{\color {red}i^{2}}\sin ^{2}\theta +i^{3}\sin ^{3}\theta \\&=\cos ^{3}\theta +3\cos ^{2}\theta \,i\sin \theta +3\cos \theta \,i^{2}\sin ^{2}\theta +{\color {red}i^{3}}\sin ^{3}\theta \\&=\cos ^{3}\theta \\&=\cos ^{3}\theta +3i\cos ^{2}\theta \sin \theta \\&=\cos ^{3}\theta +3i\cos ^{2}\theta \sin \theta -3\cos \theta \sin ^{2}\theta \\&=\cos ^{3}\theta +3i\cos ^{2}\theta \sin \theta -3\cos \theta \sin ^{2}\theta -i\sin ^{3}\theta \\\cos 3\theta +i\sin 3\theta \,\,&=(\cos ^{3}\theta -3\cos \theta \sin ^{2}\theta )+i(3\cos ^{2}\theta \sin \theta -\sin ^{3}\theta )(1)\\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/119cb56937461289d6e24f8263674b623e9db143)
![{\displaystyle {\begin{alignedat}{4}\cos 3\theta \,\,&=\cos ^{3}\theta -3\cos \theta \sin ^{2}\theta \\&=\cos ^{3}\theta -3\cos \theta (1-\cos ^{2}\theta )\\&=4\cos ^{3}\theta -3\cos \theta \\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36d3bc27e90bf8bce1680263beb98eb7e0ba741d)
![{\displaystyle {\begin{alignedat}{4}\sin 3\theta \,\,&=3\cos ^{2}\theta \sin \theta -\sin ^{3}\theta \\&=3(1-\sin ^{2}\theta )\sin \theta -\sin ^{3}\theta \\&=3\sin \theta -4\sin ^{3}\theta \\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a40bb3f8342c2d0d4ec6d1759f1c69bde7e9ba4d)
Powers
![{\displaystyle a^{n}=\underbrace {a\times a\times \cdots \times a\times a} _{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/902993a81cc823af4d53aa9b357501eb5fdd0a07)
![{\displaystyle a^{1}=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7545d7ed2d371323fca6aa910f4e9a41f86cd5)
![{\displaystyle a^{2}=a\times a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2dafd1b10496b47d54522ec5c0b73da18ac2ccc)
![{\displaystyle a^{3}=a\times a\times a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5be26ff5d0fa926b769d610cccacf7767df35411)
![{\displaystyle a^{4}=a\times a\times a\times a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38b81e7ba91a3aa461e0b7b4cd63eede068546f6)
![{\displaystyle a^{5}=a\times a\times a\times a\times a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bba6eec627f399fc156b0b163b4db35f62bfae0a)
![{\displaystyle a^{n}=a^{n-1}\times a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8bbf807cc812e7581348ef26a2c183600dca7c)
![{\displaystyle a^{n-1}={\frac {a^{n}}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4739badc4514740332a472894b77c61afed117dc)
![{\displaystyle a^{0}={\frac {a^{1}}{a}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7026691d4d036a884bbdb0c03969c8b279e3d1)
![{\displaystyle a^{-1}={\frac {a^{0}}{a}}={\frac {1}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f6618f1e01b3b9fd4ab35e30277c4db99a9c26e)
![{\displaystyle a^{-2}={\frac {a^{-1}}{a}}={\frac {\frac {1}{a}}{a}}={\frac {1}{a^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85a204bc9ed367648fa22eb6d06d3e56a8799e8e)
![{\displaystyle a^{-n}={\frac {1}{a^{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c633cad77083697a0ab944d0018590fee0622f60)
Misc
- e is Euler's number, the base of natural logarithms,
- i is the imaginary unit, which satisfies i2 = −1, and
- π is pi, the ratio of the circumference of a circle to its diameter.
![{\displaystyle f(n)={\begin{cases}n/2&\quad {\text{if }}n{\text{ is even}}\\-(n+1)/2&\quad {\text{if }}n{\text{ is odd}}\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/155e62081303ce9bfb295d5b1c70a9168e391d98)
![{\displaystyle {\begin{cases}n/2&\quad {\text{if }}n{\text{ is even}}\\-(n+1)/2&\quad {\text{if }}n{\text{ is odd}}\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68c9ead67036f95d8372c766616a669d88706a69)