DERIVADADELAFUNCIÓNCOMPUESTA.REGLADELACADENA

(g ! f ) '(x) = g ' f (x)⎡⎣ ⎤⎦⋅ f '(x)

Derivadadelapotenciadeunafunción: f n (x) f n (x)( ) ' = nf n−1(x) f '(x)

Derivadadelaraízcuadradadeunafunción: f (x)

f (x)( ) ' = f '( x )2 f ( x )

Derivadadelaraízenésimadeunafunción: f (x)n

f (x)n( ) ' = f '( x )

n f n−1( x )n

Ejemplos

a) f (x) = 2x − 12

⎛

⎝⎜

⎞

⎠⎟

3

f '(x) = 3 2x − 12

⎛

⎝⎜

⎞

⎠⎟

2

⋅2

b) f (x) = 5x+ 2 f '(x) = 52 5x+ 2

c) f (x) = 2x − 44 f '(x) = 2

4 (2x − 4)34

d) f (x) = 1x= x

−12

f '(x) = − 12x−12−1= −12x−32 = −

1

2 x3

e) f (x) = x −1x+1

f '(x) =

12 x −1

(x+1)− x −1

(x+1)2=x+1− 2(x −1)2(x+1)2 x −1

=−x+3

2(x+1)2 x −1

f) f (x) = 2x f '(x) =

−2 ⋅ 12 x

x( )2

=−1xx

= −1x x

Derivadadelafunciónexponencialdebasee:e f ( x ) e f ( x )( ) ' = e f ( x ) f '(x)

Derivadadelafunciónexponencial:a f ( x ) a f ( x )( ) ' = a f ( x ) f '(x)lna Derivadadeunlogaritmoneperiano:ln f (x)

ln f (x)( ) ' = f '(x)f (x)

Derivadadeunlogaritmo:loga f (x)

loga f (x)( ) ' = f '(x)f (x)⋅ loga e

Ejemplos

a) f (x) = 2x2−1

f '(x) = 2x ⋅2x2−1 ⋅ ln 2

b) f (x) = 3 x2−1

f '(x) = 2x

2 x2 −13 x2−1 ⋅ ln3= x

x2 −13 x2−1 ⋅ ln3

c) f (x) = e1x f '(x) = − 1

x2e1x

d) f (x) = x3 ⋅e−3x f '(x) = 3x2 ⋅e−3x + x3 ⋅ (−3) ⋅e−3x = 3x2 ⋅e−3x (1− x)

e) f (x) = e2x

x

f '(x) =2 ⋅e2x ⋅ x − e2x ⋅ 1

2 x

x( )2

=

4xe2x − e2x

2 xx

=4xe2x − e2x

2x x=e2x (4x −1)2x x

f) f (x) = log2(x4 −3x) f '(x) = 4x3 −3

(x4 −3x)⋅ log2 e

g) f (x) = log4 3x3

f '(x) = 1

3 log4 3x( )2

3

⋅33x⋅ log4 =

log4 e

3x (log4 3x)23

h) f (x) = ln 1− x1+ x⎛

⎝⎜

⎞

⎠⎟= ln(1− x)− ln(1+ x)

f '(x) = −11− x

−11+ x

=−1− x −1+ x(1− x)(1+ x)

=−21− x2

i) f (x) = x5 ⋅ ln x

f '(x) = 5x4 ⋅ ln x+ x5 ⋅ 1x= 5x4 ⋅ ln x+ x4 = x4 (5ln x+1)

j) f (x) = ln5(3x) = ln3x( )5

f '(x) = 5 ln3x( )4⋅33x

=5x⋅ ln4(3x)

Derivadadelsenodeunafunción:senf (x) senf (x)( ) ' = cos f (x) f '(x)

Derivadadelcosenodeunafunción:cos f (x) cos f (x)( ) ' = −senf (x) f '(x)

Derivadadelatangentedeunafunción:tgf (x)

tgf (x)( ) ' = f '(x)cos2 f (x)

= (1+ tg 2 f (x)) f '(x)

Derivadadelarcosenodeunafunción:arcsenf (x)

arcsenf (x)( ) ' = f '(x)

1− f 2 (x)

Derivadadelarcocosenodeunafunción:arccos f (x)

arccos f (x)( ) ' = f '(x)

1− f 2 (x)

Derivadadelaarcotangentedeunafunción:arctgf (x)

arctgf (x)( ) ' = f '(x)1+ f 2 (x)

Ejemplos

a) f (x) = sen4x f '(x) = 4cos4x b) f (x) = senx4 f '(x) = 4x3 cos x4

c) f (x) = cos x5

f '(x) = −15senx

d) f (x) = cos(3x2 + x −1) f '(x) = −(6x+1)sen(3x2 + x −1)

e) f (x) = 12cos2 5x = 1

2(cos5x)2

f '(x) = 12⋅2 ⋅cos5x ⋅ (−sen5x) ⋅5= −5cos5x ⋅ sen5x

f) f (x) = tg x

f '(x) = 1

2 x⋅

1

cos2 x=

1

2 x ⋅cos2 x

g) f (x) = arcsen(2x −3) f '(x) = 2

1− (2x −3)2

h) f (x) = arctg3x2 f '(x) = 6x1+9x4

i) f (x) = arccos x2

f '(x) = − 2x

1− (x2 )2= −

2x

1− x4

EJERCICIOS

a) f (x) = x2 −1x2 −1

⎛

⎝⎜

⎞

⎠⎟

3

b) f (x) = cos3x c) f (x) = tg(ln x)

d) f (x) = sen(senx)

e) f (x) = sen ln(1−3x)

SOLUCIONES

a) f '(x) = 3 x2 −1x2 −1

⎛

⎝⎜

⎞

⎠⎟

22x(x2 −1)− (x2 −1)2x

(x2 −1)2= 3 x

2 −1x2 −1

⎛

⎝⎜

⎞

⎠⎟

2−4x

(x2 −1)2

b) f '(x) = −3x ln3sen3x

c) f '(x) = 1cos2(ln x)

⋅1x=

1xcos2(ln x)

=1xsec2(ln x) = 1

x(1+ tg 2 ln x)

d) f '(x) = cos(senx) ⋅cos x

e) f '(x) = cos ln(1−3x) ⋅ 12 ln(1−3x)

⋅1

(1−3x)⋅ (−3)