derivadas
DESCRIPTION
Formulas de DerivacionTRANSCRIPT
Tablas de derivadas
Derivadas de las funciones elementalesFuncion Funcion simple Ejemplo Funcion compuesta Ejemplo
Simple f(x) = k f(x) = 3 f(x) = k f(x) = −5f ′(x) = 0 f ′(x) = 0 f ′(x) = 0 f ′(x) = 0
Identidad f(x) = x f(x) = x f(x) = x f(x) = xf ′(x) = 1 f ′(x) = 1 f ′(x) = 1 f ′(x) = 1
Potencial f(x) = xa f(x) = x3 fa(x) (x + 1)4
f ′(x) = axa−1 f ′(x) = 3x2 afa−1(x)f ′(x) 4(x + 1)3
Irracional f(x) = n√
x f(x) = 3√
x n√
f(x) 5√
(x + 1)3f ′(x) = 1
nn√
xn−1 f ′(x) = 1
33√
x21
n n√
fn−1(x)f ′(x) 1
5 5√
[(x+1)3]43(x + 1)2
Exponencial
f(x) = ex f(x) = ex ef(x) ex3+x
f ′(x) = ex f ′(x) = ex ef(x)f ′(x) ex3+x(3x2 + 1)f(x) = ax f(x) = 5x af(x) 7x3+2x
f ′(x) = ax ln a f ′(x) = 5x ln 5 af(x) · ln a · f ′(x) 7x3+2x · ln 7 · (3x2 + 2)
Logarıtmica
f(x) = ln x f(x) = ln x ln f(x) ln(x4 + 3x2)f ′(x) = 1
x f ′(x) = 1x
1f(x)f
′(x) 1x4+3x2 (4x3 + 6x)
f(x) = loga x f(x) = log2 x loga f(x) log3(4√
x + 3)f ′(x) = 1
x·ln a f ′(x) = 1x·ln 2
1f(x)·ln af ′(x) 1
4√x+3·ln 3(4 4
√(x + 3)3)
Funcion DerivadaExponencial f(x)g(x) f(x)g(x) · ln f(x) · g′(x) + g(x) · f(x)g(x)−1 · f ′(x)
potencial (x3 + x2)ln x5(x3 + x2)ln x5 · ln(x3 + x2) · 1
x5 5x4 + ln x5 · (x3 + x2)(ln x5)−1 · (3x2 + 2x)
Funcion Funcion simple Ejemplo Funcion compuesta Ejemplo
Seno f(x) = sen(x) f(x) = sen(x) sen(f(x)) sen((x + 1)3)f ′(x) = cos(x) f ′(x) = cos(x) cos(f(x)) · f ′(x) cos((x + 1)3) · 3(x + 1)2
Coseno f(x) = cos(x) f(x) = cos(x) cos(f(x)) cos((x + 1)3)f ′(x) = −sen(x) f ′(x) = −sen(x) −sen(f(x)) · f ′(x) −sen((x + 1)3) · 3(x + 1)2
Tangentef(x) = tg(x) f(x) = tg(x) tg(f(x)) tg((x + 1)3)
f ′(x) = 1 + tg2(x) = f ′(x) = 1 + tg2(x) = (1 + tg2(f(x))) · f ′(x) = (1 + tg2((x + 1)3)) · 3(x + 1)2 == 1
cos2(x) = 1cos2(x) = 1
cos2f(x)f′(x) = 1
cos2((x+1)3)3(x + 1)2
Cotangentef(x) = cotg(x) f(x) = cotg(x) cog(f(x)) cotg(x3 + x)
f ′(x) = −1− cotg2(x) = f ′(x) = −1− cotg2(x) = (−1− cotg(f(x)))f ′(x) = (−1− cotg(x3 + x))(3x2 + 1) == −1
sen2(x) = −1sen2(x) = −1
sen2(f(x))f′(x) = −1
sen2(x3+x) (3x2 + 1)
Arco seno f(x) = arcsen(x) f(x) = arcsen(x) arcsen(f(x)) arcsen((x + 1)3)f ′(x) = 1√
1−x2 f ′(x) = 1√1−x2
1√1−f2(x)
f ′(x) 1√1−((x+1)3)2
3(x + 1)2
Arco coseno f(x) = arccos(x) f(x) = arccos(x) arccos(f(x)) arccos((x + 1)3)f ′(x) = −1√
1−x2 f ′(x) = −1√1−x2
−1√1−f2(x)
f ′(x) −1√1−((x+1)3)2
3(x + 1)2
Arco tangente f(x) = arctg(x) f(x) = arctg(x) arctg(f(x)) arctg(x4)f ′(x) = 1
1+x2 f ′(x) = 11+x2
11+f2(x)f
′(x) 11+(x4)2 4x3
1
2
Derivadas de las funciones elementalesFuncion Derivada
Suma f(x) + g(x) f ′(x) + g′(x)ln x2 + sen(x) 1
x2 2x + cos(x)
Producto f(x) · g(x) f ′(x) · g(x) + f(x) · g′(x)ln x2 · sen(x) 1
x2 2x · sen(x) + ln x2 · cos(x)
Cocientef(x)g(x)
f ′(x)·g(x)−f(x)·g′(x)g2(x)
ln x2
sen(x)
1x2 2x·sen(x)−ln x2·cos(x)
sen2(x)
Compuesta g(f(x)) g′(f(x)) · f ′(x)ln(sen(x)) 1
sen(x) · cos(x)