AP Calculus Derivative and Integral Formulas

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(d/dx) tan x

sec²x

∫cos x dx

sin x + C

∫ sec² x dx

tan x + C

(d/dx) f(u)

u' f'(u)

(d/dx) ln u

u'/u

(d/dx) (uv)

uv' + vu'

(d/dx) sec x

sec x tan x

∫xⁿ dx

( [xⁿ⁺¹]/[n+1] ) + C, n≠-1

∫ [ (u')/ (a²+u²) ] du

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

(d/dx) t/b

(bt' - tb')/(b²)

∫ tan x dx

- ln |cosx| +C

∫sin x dx

-cos x + C

∫ csc² x dx

-cot x + C

∫csc x cot x dx

-csc x + C

(d/dx) cotx

-csc²x

∫cot x dx

-ln |csc x| + C

(d/dx) cos x

-sin x

∫ sec x tan x dx

sec x + C

(d/dx) log base a of x

1/ x ln a

(d/dx) arctan x

1/(1+x²)

(d/dx) ln x

1/x

(d/dx) arcsin x

1/√(1-x²)

A = (1/2) w (h1 + h2)

Area of Trapezoid

(slope between two points) [f(b)-f(a)]/[b-a]

Average Rate of Change

f ' (c)

Instant Rate of Change

a function f that is continuous on [a,b] takes on every y-value between f(a) and f(b)

Intermediate Value Theorem

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

Mean Value Theorem

∫ 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(b) = f(a) + ∫ f ' (x) dx

Start Plus Net Change/Accumulation

Volume of a Washer

V = π ∫ (R²-r²) dx

Volume of a disc

V = π∫r² dx

Volume of the Cross Section

V = ∫ A dx

∫a^x dx

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

(d/dx) a^x

a^x ln a

∫[ (u')/ (√a²-u²) ] du

arcsin (u/a) + C

(d/dx) sin x

cos x

(d/dx) e^x

e^x

∫ e^x dx

e^x + C

∫tan x dx

ln |secx| + C

∫ cot x dx

ln |sinx| + C

∫ (u'/u) dx

ln |u| + C

∫ 1/x dx

ln |x| + C

(d/dx) xⁿ

nxⁿ⁻¹

speed =

|v(t)|

Total distance

∫ |v(t)|dt


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