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)
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