Tabla de derivadas Reglas generales de derivación 1 Regla de la suma-resta f (x) = u(x) ± v(x) f 0 (x) = u0 (x) ± v 0 (x) Regla del producto1 f (x) = u(x) · v(x) f 0 (x) = u0 (x) · v(x) + u(x) · v 0 (x) Regla del cociente f (x) = Regla de la cadena f (x) = (u ◦ v)(x) = u(v(x)) f 0 (x) = u0 (v(x)) · v 0 (x) Inversa de una función f (x) = u−1 (x) f 0 (x) = u(x) v(x) f 0 (x) = u0 (x)·v(x)−u(x)·v 0 (x) v(x)2 1 (u0 ◦ u−1 )(x) = 1 u0 (u−1 (x)) En general, si f (x) = u1 (x) · u2 (x) · . . . · un (x) entonces f 0 (x) = u01 (x) · u2 (x) · . . . · un (x) + u1 (x) · u02 (x) · . . . · un (x) + · · · + u1 (x) · u2 (x) · . . . · u0n (x). Tipo f (x) f 0 (x) Constante f (x) = k f 0 (x) = 0 Identidad f (x) = x f 0 (x) = 1 Potencial f (x) = xn f 0 (x) = n xn−1 Irracional f (x) = Exponencial f (x) = ex f 0 (x) = ex Exponencial en base a f (x) = ax f 0 (x) = ax · log a Exponencial de funciones f (x) = g(x)h(x) con a > 0 0 (x) f 0 (x) = g(x)h(x) · h0 (x) · log g(x) + h(x) · g g(x) Logarítmica f (x) = log x f 0 (x) = 1 x Logarítmica en base a f (x) = loga x f 0 (x) = 1 x·log a Seno f (x) = sin x f 0 (x) = cos x Coseno f (x) = cos x f 0 (x) = − sin x √ n f 0 (x) = x .. . 1 Restricciones n 1 √ n n−1 x con a > 0, a 6= 1 Tipo f (x) f 0 (x) Tangente f (x) = tan x f 0 (x) = 1 + tan2 x = Cosecante f (x) = csc x = 1 sin x f 0 (x) = − csc x · cot x Secante f (x) = sec x = 1 cos x f 0 (x) = sec x · tan x Cotangente f (x) = cot x = 1 tan x f 0 (x) = − csc2 x = Arco seno f (x) = arc sin x f 0 (x) = √ 1 1−x2 Arco coseno f (x) = arc cos x f 0 (x) = √ −1 1−x2 Arco tangente f (x) = arc tan x f 0 (x) = 1 1+x2 Arco cosecante f (x) = arc csc x f 0 (x) = √−1 x x2 −1 Arco secante f (x) = arc sec x f 0 (x) = √1 x x2 −1 Arco cotangente f (x) = arc cot x f 0 (x) = −1 1+x2 Seno hiperbólico f (x) = sinh x f 0 (x) = cosh x Coseno hiperbólico f (x) = cosh x f 0 (x) = sinh x Tangente hiperbólico f (x) = tanh x f 0 (x) = Cosecante hiperbólico f (x) = csch x = 1 sinh x f 0 (x) = − csch x · coth x Secante hiperbólico f (x) = sech x = 1 cosh x f 0 (x) = − sech x · tanh x Cotangente hiperbólico f (x) = coth x = 1 tanh x f 0 (x) = − csch2 x = Arco seno hiperbólico f (x) = arc sinh x f 0 (x) = √ 1 x2 +1 Arco coseno hiperbólico f (x) = arc cosh x f 0 (x) = √ 1 x2 −1 Arco tangente hiperbólico f (x) = arc tanh x f 0 (x) = 1 1−x2 Arco cosecante hiperbólico f (x) = arc csch x f 0 (x) = √−1 |x| 1+x2 Arco secante hiperbólico f (x) = arc sech x f 0 (x) = √−1 |x| 1−x2 Arco cotangente hiperbólico f (x) = arc coth x f 0 (x) = −1 x2 −1 Restricciones 1 cos2 x −1 sin2 x 1 cosh2 x −1 sinh2 x Usando la regla de la cadena se obtiene una tabla similar a la anterior para funciones compuestas. Por ejemplo, Si f (x) = log v(x), entonces f 0 (x) = 2 v 0 (x) . v(x)