Calculus

\frac{d}{d x} [c u] = c u ´

\frac{d}{d x} [u \pm v] = u ´ \pm v ´

\frac{d}{d x} [u v] = u v ´ + v u ´

\frac{d}{d x} [\frac{u}{v}] = \frac{v u ´ - u v ´}{v^{2}}

\frac{d}{d x} [c] = 0

\frac{d}{d x} [u^{n}] = n u^{n - 1} u ´

\frac{d}{d x} [x] = 1

\frac{d}{d x} [| u |] = \frac{u}{| u |} (u ´), u \neq 0

\frac{d}{d x} [\ln u] = \frac{u ´}{u}

\frac{d}{d x} [e^{u}] = e^{u} u ´

\frac{d}{d x} [\log_{a} u] = \frac{u ´}{(\ln a) u}

\frac{d}{d x} [a^{u}] = (\ln a) a^{u} u ´

\frac{d}{d x} [\sin u] = (\cos u) u ´

\frac{d}{d x} [\cos u] = - (\sin u) u ´

\frac{d}{d x} [\tan u] = (\sec^{2} u) u ´

\frac{d}{d x} [\cot u] = - (\csc^{2} u) u ´

\frac{d}{d x} [\sec u] = (\sec u \tan u) u ´

\frac{d}{d x} [\csc u] = - (\csc u \cot u) u ´

\frac{d}{d x} [arc \sin u] = \frac{u ´}{\sqrt{1 - u^{2}}}

\frac{d}{d x} [arc \cos u] = \frac{- u ´}{\sqrt{1 - u^{2}}}

\frac{d}{d x} [arc \tan u] = \frac{u ´}{1 + u^{2}}

\frac{d}{d x} [arc \cot u] = \frac{- u ´}{1 + u^{2}}

\frac{d}{d x} [arc \sec u] = \frac{u ´}{| u | \sqrt{u^{2} - 1}}

\frac{d}{d x} [arc \csc u] = \frac{- u ´}{| u | \sqrt{u^{2} - 1}}

\int k f (u) d u = k \int f (u) d u

\int [f (u) \pm g (u)] d u = \int f (u) d u \pm \int g (u) d u

\int d u = u + C

\int u^{n} d u = \frac{u^{n + 1}}{n + 1} + C, n \neq - 1

\int \frac{d u}{u} = \ln | u | + C

\int e^{u} d u = e^{u} + C

\int a^{u} d u = (\frac{1}{\ln a}) a^{u} + C

\int \sin u d u = - \cos u + C

\int \cos u d u = \sin u + C

\int \tan u d u = - \ln | \cos u | + C

\int \cot u d u = \ln | \sin u | + C

\int \sec u d u = \ln | \sec u + \tan u | + C

\int \csc u d u = - \ln | \csc u + \cot u | + C

\int \sec^{2} u d u = \tan u + C

\int \csc^{2} u d u = - \cot u + C

\int \sec u \tan u d u = \sec u + C

\int \csc u \cot u d u = - \csc u + C

\int \frac{d u}{\sqrt{a^{2} - u^{2}}} = \arcsin \frac{u}{a} + C

\int \frac{d u}{a^{2} + u^{2}} = \frac{1}{a} \arctan \frac{u}{a} + C

\int \frac{d u}{u \sqrt{u^{2} - a^{2}}} = \frac{1}{a} arcsec \frac{| u |}{a} + C