Calculus

Basic Differentiation Rules

ddx[cu]=cu´ddx[cu]=cu´
ddx[u±v]=u´±v´ddx[u±v]=u´±v´
ddx[uv]=uv´+vu´ddx[uv]=uv´+vu´
ddx[uv]=vu´+uv´v2ddx[uv]=vu´+uv´v2
ddx[c]=0ddx[c]=0
ddx[un]=nun-1u´ddx[un]=nun1u´
ddx[x]=1ddx[x]=1
ddx[|u|]=u|u|(u´), u0ddx[|u|]=u|u|(u´), u0
ddx[lnu]=u´uddx[lnu]=u´u
ddx[eu]=euu´ddx[eu]=euu´
ddx[logau]=u´(lna)uddx[logau]=u´(lna)u
ddx[au]=(lna)auu´ddx[au]=(lna)auu´
ddx[sinu]=(cosu)u´ddx[sinu]=(cosu)u´
ddx[cosu]=-(sinu)u´ddx[cosu]=(sinu)u´
ddx[tan u]=(sec2 u)u´ddx[tan u]=(sec2 u)u´
ddx[cot u]=-(csc2 u)u´ddx[cot u]=(csc2 u)u´
ddx[sec u]=(sec u tan u)u´ddx[sec u]=(sec u tan u)u´
ddx[csc u]=-(csc u cot u)u´ddx[csc u]=(csc u cot u)u´
ddx[arcsin u]=u´1-u2ddx[arcsin u]=u´1u2
ddx[arc cos u]=-u´1-u2ddx[arc cos u]=u´1u2
ddx[arc tan u]=u´1+u2ddx[arc tan u]=u´1+u2
ddx[arc cot u]=-u´1+u2ddx[arc cot u]=u´1+u2
ddx[arc sec u]=u´|u|u2-1ddx[arc sec u]=u´|u|u21
ddx[arc csc u]=-u´|u|u2-1ddx[arc csc u]=u´|u|u21

Basic Integration Rules (a > 0)

k f(u)du=kf(u)du
[f(u)±g(u)]du=f(u)du ±g(u)du
du=u+C
undu=un+1n+1+C, n-1
duu=ln|u|+C
eudu=eu+C
audu=(1lna)au+C
sinu du=-cosu+C
cosu du=sinu+C
tanu du=-ln|cosu|+C
cotu du=ln|sinu|+C
secu du=ln|secu+tanu|+C
cscu du=-ln|cscu +cotu| + C
secu du =ln|secu+tanu|+C
cscu du =-ln|cscu+cotu|+C
sec2u du =tanu+C
csc2u du =-cotu+C
secu tan u du =secu+C
csc u cot u du =-csc u+C
du2a2-u2= arcsin ua+ C
dua2+u2= 1aarctan ua+ C
duu2u2-a2= 1aarcsec |u|a+ C
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