Derivatives and Limits: Rules, Properties, and Common Identities

Derivadas

derivative rules

der(x^a) = a . x^a-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( e^x) = e^x

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

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

common limits

lim_x->cf(x) = ∞

lim_x->cg(x) = L

lim_x->c[f(x) +- g(x)] = ∞

lim_x->c[f(x) . g(x)] = ∞, L>0

lim_x->c[f(x) . g(x)] = - ∞, L<0

lim_x->∞(1 + k/x)^x = e^k

lim_x->∞[axⁿ] = ∞, 0

lim_x->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

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

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

lim_x->0[1+x]^1/x = e

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

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

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

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

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

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

lim_x->c[a^x] = a^c, a > 0