Derivatives and Integrals Basic Rules AP Calculus AB

(d/dx) sin x

cos x

(d/dx) cos x

-sin x

(d/dx) tan x

sec²x

(d/dx) cotx

-csc²x

(d/dx) sec x

sec x tan x

(d/dx) arcsin x

1/√(1-x²)

(d/dx) arctan x

1/(1+x²)

(d/dx) xⁿ

nxⁿ⁻¹

(d/dx) ln x

1/x

(d/dx) ln u

u'/u

(d/dx) log base a of x

1/ x ln a

(d/dx) e^x

e^x

(d/dx) a^x

a^x ln a

(d/dx) f(u)

u' f'(u)

(d/dx) t/b

(bt' - tb')/(b²)

(d/dx) (uv)

uv' + vu'

∫ 1/x dx

ln |x| + C

∫ (u'/u) dx

ln |u| + C

∫ e^x dx

e^x + C

∫a^x dx

\[(a^x)/ln a] + C

∫cos x dx

sin x + C

∫sin x dx

-cos x + C

∫ sec² x dx

tan x + C

∫ tan x dx

ln |cosx| +C

∫ cot x dx

ln |sinx| + C

∫ sec x tan x dx

sec x + C

∫ csc² x dx

-cot x + C

∫csc x cot x dx

-csc x + C

∫ [ (u')/ (a²+u²) ] du

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

∫[ (u')/ (√a²-u²) ] du

arcsin (u/a) + C

∫tan x dx

ln |secx| + C

∫cot x dx

-ln |csc x| + C

∫ f ' (x) dx \= f(b) - f(a)

Net Change/ Fundamental Theorem of Calculus

d/dx ∫ f(t) dt \= f(v)v' - f(u)u'

Second Fundamental Theorem of Calculus

f ' (c) \= f(b) - f(a) / (b-a)

Mean Value Theorem

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(-1+u^2))

d/dx\[arccsc u)

\-u’/(|u|\*sqrt(-1+u^2))

∫ du/(sqrt((a^2)-(u^2))

arcsin(u/a)+C

∫ du/(u(sqrt((u^2)-(a^2))

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

∫ du/((u^2)+(a^2))

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