Lista Derivadas PDF

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Es una lista de las derivadas trigonométricas,hiperbólicas......

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E I I – UVa. Departamento de M atem´ atica Aplicada (RMdeF)

Curso 2018-19

1

´ REGLAS DE DERIVACION [ ]′ [ ]′ [ ]′ • f (x) ± g(x) = f ′ (x) ± g ′ (x) • f (x) · g (x) = f ′ (x) g (x) + f (x) g ′ (x) • k·f (x) = k f ′ (x) [ ]′ ] [ −g ′ (x) f ′ (x) g(x) − f (x) g ′ (x) f (x) ′ 1 • = = • 2 g(x) g(x) [g(x)] [g(x)]2 [ ]′ ( ) • Regla de la cadena: g(f (x) = g ′ f (x) · f ′ (x) ; o bien, expresado de otro modo: [g (u)]′ = u′ · g ′ (u). Funciones b´ asicas: ′

Funci´ on compuesta:

′

(k) = 0;

(x) = 1

′

(sen u) = u′ cos u

′

′

(cos u) = −u′ sen u

(sen x) = cos x

′

(cos x) = − sen x ′

(tg x) =

1 = 1 + tg2 x cos2 x

(eu ) = u′ eu

′

′

(ln u) = u′ / u

(ln x) = 1/x ′

′

(ax ) = ax ln a ′

u′ = u′ (1 + tg2 u) cos2 u

′

′

(ex ) = ex

(loga x) =

′

(tg u) =

(au ) = u′ au ln a

1 x ln a

′

(loga u) =

′

′

(xa ) = a xa−1 )′ √ ′ ( ( n x ) = x1/n =

u′ u ln a

(ua ) = a ua−1 u′ n

√ n

1 x n−1

u′ √ n n u n−1

√ ′ (n u) =

′

(sh u) = u′ ch u

′

(ch u) = u′ sh u

(sh x) = ch x (ch x) = sh x 1 = 1−tgh2 x ch 2 x 1 ′ (arc sen x) = √ 1 − x2 −1 ′ (arc cos x) = √ 1 − x2 1 ′ (arc tg x) = 1 + x2 1 ′ (arg sh x) = √ 2 x +1 1 ′ (arg ch x) = √ 2 x −1 1 ′ (arg th x) = 1 − x2 ′

(tgh x) =

′

′

u′

′

(tgh u) =

ch 2 u ′

(arc sen u) = √

= u′ (1 − tgh2 u) u′

1 − u2 −u′ ′ (arc cos u) = √ 1 − u2 ′ u ′ (arc tg u) = 1 + u2 u′ ′ (arg sh u) = √ u2 + 1 u′ ′ (arg ch u) = √ u2 − 1 u′ ′ (arg th u) = 1 − u2

Nota: Las funciones b´ a sicas son derivables en todos los puntos de su dominio   √ n     • x en a = 0. excepto: • arc sen x y arc cos x en a = −1 y a = 1 , en los cuales se tiene derivada infinita (tangente vertical).     • arg ch x en a =1...