User:Auximines/sandbox

Differentiate x squared

$f(x)=x^{2}$

${\frac {d}{dx}}(f(x))=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$

${\begin{alignedat}{4}{\frac {d}{dx}}(x^{2})\,\,&=\lim _{h\to 0}{\frac {(x+h)^{2}-x^{2}}{h}}\\&=\lim _{h\to 0}{\frac {x^{2}+2xh+h^{2}-x^{2}}{h}}\\&=\lim _{h\to 0}{\frac {2xh+h^{2}}{h}}\\&=\lim _{h\to 0}2x+h\\&=2x\end{alignedat}}$

Differentiate x cubed

$f(x)=x^{3}$

${\frac {d}{dx}}(f(x))=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$

${\begin{alignedat}{4}{\frac {d}{dx}}(x^{3})\,\,&=\lim _{h\to 0}{\frac {(x+h)^{3}-x^{3}}{h}}\\&=\lim _{h\to 0}{\frac {x^{3}+3x^{2}h+3xh^{2}+h^{3}-x^{3}}{h}}\\&=\lim _{h\to 0}{\frac {3x^{2}h+3xh^{2}+h^{3}}{h}}\\&=\lim _{h\to 0}3x^{2}+3xh+h^{2}\\&=3x^{2}\end{alignedat}}$

Differentiate x^n

$f(x)=x^{n}$

${\frac {d}{dx}}(f(x))=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$

${\begin{alignedat}{4}{\frac {d}{dx}}(x^{n})\,\,&=\lim _{h\to 0}{\frac {(x+h)^{n}-x^{n}}{h}}\\&=\lim _{h\to 0}{\frac {x^{n}+nx^{n-1}h+{\frac {n(n-1)}{2!}}x^{n-2}h^{2}+{\frac {n(n-1)(n-2)}{3!}}x^{n-3}h^{3}+\,...-x^{n}}{h}}\\&=\lim _{h\to 0}{\frac {nx^{n-1}h+{\frac {n(n-1)}{2!}}x^{n-2}h^{2}+{\frac {n(n-1)(n-2)}{3!}}x^{n-3}h^{3}+\,...}{h}}\\&=\lim _{h\to 0}nx^{n-1}+{\frac {n(n-1)}{2!}}x^{n-2}h+{\frac {n(n-1)(n-2)}{3!}}x^{n-3}h^{2}+\,...\\&=\lim _{h\to 0}nx^{n-1}\\{\frac {d}{dx}}(x^{n})\,\,&=nx^{n-1}\end{alignedat}}$

Differentiate sin x

$f(x)=\sin x$

${\frac {d}{dx}}(f(x))=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$

$\lim _{h\to 0}{\frac {\sin h}{h}}=1$

$\lim _{h\to 0}{\frac {1-\cos h}{h}}=0$

$\sin(a+b)=\sin a\cos b+\cos a\sin b$

${\begin{alignedat}{3}{\frac {d}{dx}}(\sin x)\,\,&=\lim _{h\to 0}{\frac {\sin(x+h)-\sin x}{h}}\\&=\lim _{h\to 0}{\frac {\sin x\cos h+\cos x\sin h-\sin x}{h}}\\&=\lim _{h\to 0}\sin x{\frac {\cos h-1}{h}}+\lim _{h\to 0}\cos x{\frac {\sin h}{h}}\\\end{alignedat}}$

Multiple Angle

${\begin{cases}\sin 2\theta =2\sin \theta \cos \theta \\\cos 2\theta =2\cos ^{2}\theta -1\\\end{cases}}$

$\cos 3\theta =?$

$\sin 4\theta =?$

$\cos 12\theta =?$

$e^{ix}=\cos x+i\sin x$

$e^{3i\theta }$

$e^{3i\theta }={(e^{i\theta })}^{3}$

${\begin{alignedat}{4}\cos 3\theta +i\sin 3\theta \,\,&=\\\cos 3\theta +i\sin 3\theta \,\,&=(\cos \theta +i\sin \theta )^{3}\\&=\cos ^{3}\theta \\&=\cos ^{3}\theta +3\cos ^{2}\theta \,i\sin \theta \\&=\cos ^{3}\theta +3\cos ^{2}\theta \,i\sin \theta +3\cos \theta \,i^{2}\sin ^{2}\theta \\&=\cos ^{3}\theta +3\cos ^{2}\theta \,i\sin \theta +3\cos \theta \,i^{2}\sin ^{2}\theta +i^{3}\sin ^{3}\theta \\&=\cos ^{3}\theta +3\cos ^{2}\theta \,{\color {red}i}\sin \theta +3\cos \theta \,i^{2}\sin ^{2}\theta +i^{3}\sin ^{3}\theta \\&=\cos ^{3}\theta +3\cos ^{2}\theta \,i\sin \theta +3\cos \theta \,{\color {red}i^{2}}\sin ^{2}\theta +i^{3}\sin ^{3}\theta \\&=\cos ^{3}\theta +3\cos ^{2}\theta \,i\sin \theta +3\cos \theta \,i^{2}\sin ^{2}\theta +{\color {red}i^{3}}\sin ^{3}\theta \\&=\cos ^{3}\theta \\&=\cos ^{3}\theta +3i\cos ^{2}\theta \sin \theta \\&=\cos ^{3}\theta +3i\cos ^{2}\theta \sin \theta -3\cos \theta \sin ^{2}\theta \\&=\cos ^{3}\theta +3i\cos ^{2}\theta \sin \theta -3\cos \theta \sin ^{2}\theta -i\sin ^{3}\theta \\\cos 3\theta +i\sin 3\theta \,\,&=(\cos ^{3}\theta -3\cos \theta \sin ^{2}\theta )+i(3\cos ^{2}\theta \sin \theta -\sin ^{3}\theta )(1)\\\end{alignedat}}$

${\begin{alignedat}{4}\cos 3\theta \,\,&=\cos ^{3}\theta -3\cos \theta \sin ^{2}\theta \\&=\cos ^{3}\theta -3\cos \theta (1-\cos ^{2}\theta )\\&=4\cos ^{3}\theta -3\cos \theta \\\end{alignedat}}$

${\begin{alignedat}{4}\sin 3\theta \,\,&=3\cos ^{2}\theta \sin \theta -\sin ^{3}\theta \\&=3(1-\sin ^{2}\theta )\sin \theta -\sin ^{3}\theta \\&=3\sin \theta -4\sin ^{3}\theta \\\end{alignedat}}$

${\begin{cases}\cos 3\theta =4\cos ^{3}\theta -3\cos \theta \\\sin 3\theta =3\sin \theta -4\sin ^{3}\theta \\\end{cases}}$

Powers

$a^{n}=\underbrace {a\times a\times \cdots \times a\times a} _{n}$

$a^{1}=a$

$a^{2}=a\times a$

$a^{3}=a\times a\times a$

$a^{4}=a\times a\times a\times a$

$a^{5}=a\times a\times a\times a\times a$

$a^{n}=a^{n-1}\times a$

$a^{n-1}={\frac {a^{n}}{a}}$

$a^{0}={\frac {a^{1}}{a}}=1$

$a^{-1}={\frac {a^{0}}{a}}={\frac {1}{a}}$

$a^{-2}={\frac {a^{-1}}{a}}={\frac {\frac {1}{a}}{a}}={\frac {1}{a^{2}}}$

$a^{-n}={\frac {1}{a^{n}}}$

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.

$f(n)={\begin{cases}n/2&\quad {\text{if }}n{\text{ is even}}\\-(n+1)/2&\quad {\text{if }}n{\text{ is odd}}\\\end{cases}}$

${\begin{cases}n/2&\quad {\text{if }}n{\text{ is even}}\\-(n+1)/2&\quad {\text{if }}n{\text{ is odd}}\\\end{cases}}$