# Derivatives and Limits: Rules, Properties, and Common Identities

Classified in Greek

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### derivative rules

der(xa) = a . xa-1

der(a) = 0

(f+-g)' = f' + g'

(a.f)' = a .f'

(f/g)' = ((f'.g-g'.f)/g2)

der( ln(x) ) = 1/x

der( ln|x| ) = 1/x

der( ex) = ex

der( log(x) ) = 1/(x.ln(10))

der( loga(x) ) = 1/(x.ln(a))

### common limits

limx->cf(x) = ∞

limx->cg(x) = L

limx->c[f(x) +- g(x)] = ∞

limx->c[f(x) . g(x)] = ∞, L>0

limx->c[f(x) . g(x)] = - ∞, L<0

limx->∞(1 + k/x)x = ek

limx->∞[axn] = ∞, 0

limx->0(1 +x)1/x = e

## Trigonometria

### basic identities

tan(x) = sin(x)/cos(x)

cot(x) = cos(x)/sin(x)

sec(x) = 1/cos(x)

csc(x) = 1/sin(x)

### pythagorean identities

sin(2x) = 2 sin(x)cos(x)

cos(2x) = 1 - 2.sin2(x)

tan(2x) = 2tan(x) / 1-tan2(x)

sin(x) | -1<= y <= 1 | arcsin(x) | - (π/2) <= y <= π/2

cos(x) | -1<= y <= 1 | arccos(x) | 0 <= y <= π

## Limites

### three special limits

limx->0[sin(x)/x] = 1

limx->0[(1 - cos(x)) /x] = 0

limx->0[1+x]1/x = e

limx->c[sin(x)] = sin c

limx->c[cos(x)] = cos c

limx->c[tan(x)] = tan c

limx->c[cot(x)] = cot c

limx->c[sec(x)] = sec c

limx->c[csc(x)] = csc c

limx->c[ax] = ac, a > 0