derivadas

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)