MTH 267 - Basic Derivatives and Integrals

Flashcards covering basic derivatives and integrals. Note: For deratiives u is u(x) & v is v(x) - apply chain rule when necessary.

d/dx [cu] =

cu'

d/dx [u ± v] =

u' ± v'

d/dx [uv] =

uv' + vu'

d/dx [u/v] =

(vu' - uv') / v^2

d/dx [c] =

0

d/dx [u^n] =

nun^(n-1)u'

d/dx [x] =

1

d/dx [|u|] =

(u' * u) / |u|, u != 0

d/dx [ln u] =

u' / u

d/dx [e^u] =

e^u * u'

d/dx [loga u] =

u' / ((ln a) * u)

d/dx [a^u] =

(ln a) * a^u * u'

d/dx [sin u] =

(cos u) * u'

d/dx [cos u] =

-(sin u) * u'

d/dx [tan u] =

(sec^2 u) * u'

d/dx [cot u] =

-(csc^2 u) * u'

d/dx [sec u] =

(sec u tan u) * u'

d/dx [csc u] =

-(csc u cot u) * u'

d/dx [arcsin u] =

u' / sqrt(1 - u^2)

d/dx [arccos u] =

-u' / sqrt(1 - u^2)

d/dx [arctan u] =

u' / (1 + u^2)

d/dx [arccot u] =

-u' / (1 + u^2)

d/dx [arcsec u] =

u' / (|u| * sqrt(u^2 - 1))

d/dx [arccsc u] =

-u' / (|u| * sqrt(u^2 - 1))

d/dx [sinh u] =

(cosh u) * u'

d/dx [cosh u] =

(sinh u) * u'

d/dx [tanh u] =

(sech^2 u) * u'

d/dx [coth u] =

-(csch^2 u) * u'

d/dx [sech u] =

-(sech u tanh u) * u'

d/dx [csch u] =

-(csch u coth u) * u'

d/dx [sinh^-1 u] =

u' / sqrt(u^2 + 1)

d/dx [cosh^-1 u] =

u' / sqrt(u^2 - 1)

d/dx [tanh^-1 u] =

u' / (1 - u^2)

d/dx [coth^-1 u] =

u' / (1 - u^2)

d/dx [sech^-1 u] =

-u' / (u * sqrt(1 - u^2))

d/dx [csch^-1 u] =

-u' / (|u| * sqrt(1 + u^2))

∫ du =

u + C

∫ k du =

ku + C

∫ u^n du =

(u^(n+1))/(n+1) + C, for n != -1

∫ cos u du =

sin u + C

∫ sin u du =

-cos u + C

∫ sec^2 u du =

tan u + C

∫ sec u tan u du =

sec u + C

∫ csc^2 u du =

-cot u + C

∫ csc u cot u du =

-csc u + C

∫ e^u du =

e^u + C

∫ a^u du =

(1/ln a) * a^u + C, a > 0

∫ 1/u du =

ln |u| + C

∫ tan u du =

-ln |cos u| + C = ln |sec u| + C

∫ cot u du =

ln |sin u| + C = -ln |csc u| + C

∫ sec u du =

ln |sec u + tan u| + C = -ln |sec u - tan u| + C

∫ csc u du =

-ln |csc u + cot u| + C = ln |csc u - cot u| + C

∫ du / sqrt(a^2 - u^2) =

arcsin(u/a) + C

∫ du / (a^2 + u^2) =

(1/a) * arctan(u/a) + C

∫ du / (u * sqrt(u^2 - a^2)) =

(1/a) * arcsec(|u|/a) + C

∑ (from i=1 to n) c =

cn

∑ (from i=1 to n) i =

n(n+1)/2

∑ (from i=1 to n) i^2 =

n(n+1)(2n+1)/6

∑ (from i=1 to n) i^3 =

(n^2(n+1)^2)/4

∑ (from i=1 to n) i^4 =

(n(2n+1)(n+1)(3n^2+3n-1)) / 30

Lower Sums (L.S.) =

lim n→∞ ∑ (from i=1 to n) f(mi)∆x, where f(mi) is the min. value of f on the subinterval

Upper Sums (U.S.) =

lim n→∞ ∑ (from i=1 to n) f(Mi)∆x, where f(Mi) is the max. value of f on the subinterval

Left Endpoints =

a + (i - 1)∆x, for i = 1, …, n

Right Endpoints =

a + i∆x, for i = 1, …, n

∆x =

(b - a) / n