Derivadas

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Derivadas

1 DEFINICIÓNUna función definida en un entorno de x=a, se dice que es derivable en ese punto si existe el límite:

ƒ'(a) = lim hƒ(a + h) −ƒ(a) = además, el valor de la derivada en a, f'(a), es el límite

2 EJEMPLO 1Sea f(x)=1x+9;

f(2) = 2+9 = 11f(2+h) = 1·(2+h)+9= 2+1·h+9=1·h+11

f(2+h)f(2) =1·h

ƒ'(2) = lim hƒ(2 + h) −ƒ(2) = lim h

1·h = 1

Así pues, la derivada en x=2 de f(x)=1x+9 es f'(2) = 1

3 EJEMPLO 2Sea f(x)=5x29x+2; f(6) = 18054+2 = 128f(6+h) = 5·(6+h)29·(6+h)+2= 5·(36+h2+12h) 54 9h + 2 = 128 + 5·h2 + 51·h

f(6+h)f(6) =5·h2 + 51·h

ƒ'(6) = lim hƒ(6 + h) −ƒ(6) = lim h

5·h + 51h = lim (5·h+ 51) = 51

Así pues, la derivada en x=6 de f(x)=5x29x+2 es f'(6) = 51

4 (axn)' = n·axn1 (12x9)' = 108x8

(1x9)' = 9x8

(5x4)' = 20x3

(9x6)' = 54x5

5 ( x )'=( x )'= 1/2·x = 2· x1

7x + 10x+ 6 = 7x + 10x+ 6 = 1/2· 7x + 10x+ 6 ·(14x+ 10)=

2· 7x + 10x + 6

14x + 10

h→0

h→0 h→0

h→0 h→02

h→0

√ 1/2 −1/2√

(√ 2 )' ( ( 2 )1/2) ' ( 2 )−1/2

√ 2

( ) ( )



10x + 9x+ 6 = 10x + 9x+ 6 = 1/2· 10x + 9x+ 6 ·(20x+ 9)=

2· 10x + 9x + 6

20x + 9

2x + 3x+ 5 = 2x + 3x+ 5 = 1/2· 2x + 3x+ 5 ·(4x+ 3)= 2· 2x + 3x + 5

4x + 3

11x + 5x+ 2 = 11x + 5x+ 2 = 1/2· 11x + 5x+ 2 ·(22x+ 5)=

2· 11x + 5x + 2

22x + 5

6 TRIGONOMÉTRICAS(sen(x))' = cos(x) (cos(x))' = sen(x) ( sen(f(x)) )' = cos(f(x))·f'(x) ( cos(f(x)) )' = sen(f(x))·f'(x)

( sen(13x12) )' = Cos(13x12)·( 156x11 )( Cos(12x19) )' = sen(12x19)·( 228x18 )( sen(9x2) )' = Cos(9x2)·( 18x1 )( Cos(5x9) )' = sen(5x9)·( 45x8 )( sen(10x14) )' = Cos(10x14)·( 140x13 )( Cos(3x15) )' = sen(3x15)·( 45x14 )( sen(1x19) )' = Cos(1x19)·( 19x18 )( Cos(2x10) )' = sen(2x10)·( 20x9 )

7 (f(x)n)' = nf(x)n1·f'(x) (sen14(x))' = 14·sen13(x)·cos(x)(cos14(x))' = 14·cos13(x)·sen(x)(sen16(x))' = 16·sen15(x)·cos(x)(cos20(x))' = 20·cos19(x)·sen(x)(sen10(x))' = 10·sen9(x)·cos(x)(cos4(x))' = 4·cos3(x)·sen(x)(sen5(x))' = 5·sen4(x)·cos(x)(cos16(x))' = 16·cos15(x)·sen(x)(sen9(x))' = 9·sen8(x)·cos(x)(cos13(x))' = 13·cos12(x)·sen(x)(sen5/9(x))' = 5/9·sen4/9(x)·cos(x)(cos8/2(x))' = 8/2·cos6/2(x)·sen(x)(sen9/8(x))' = 9/8·sen1/8(x)·cos(x)(cos9/7(x))' = 9/7·cos2/7(x)·sen(x)(sen7/9(x))' = 7/9·sen2/9(x)·cos(x)(cos4/3(x))' = 4/3·cos1/3(x)·sen(x)

(√ 2 )' ( ( 2 )1/2) ' ( 2 )−1/2

√ 2

(√ 2 )' ( ( 2 )1/2 )' ( 2 )−1/2√ 2

(√ 2 )' ( ( 2 )1/2) ' ( 2 )−1/2

√ 2



8 (f·g)'=f '·g + f·g'(10x7·sen(x))' = 70x6·sen(x) + 10x7·cos(x)(3x3·cos(x))' = 9x2·cos(x) 3x3·sen(x)(5·sen(x))' = 0·sen(x) + 5cos(x) = 5cos(x)(7·cos(x))' = 0·cos(x) 7sen(x) = 7sen(x)

9 (fog)'(x)=(f(g(x)))' = f'(g(x))·g'(x)(sen(6x4))' = (cos(6x4)) · 24x3

(8sen(6x2))' = 8cos(6x2) · 12x1

10 (f/g)' = (f '·gfg')/g2(tan(x))' = (sen(x)/cos(x))' = (cos(x)·cos(x)sen(x)·(sen(x)) )/cos2(x) = 1/cos2(x)( (5x+10)/(3x+6))' = ( (5·(3x+6) (5x+10)·3 ) / (3x+6)2

( (9x+9)/(6x+3))' = ( (9·(6x+3) (9x+9)·6 ) / (6x+3)2

( (10x+6)/(7x+7))' = ( (10·(7x+7) (10x+6)·7 ) / (7x+7)2

( (9x+9)/sen(x) )' = ( (9·sen(x) (9x+9)·cos(x) ) / sen2(x) ( (4x+5)/cos(x) )' = ( (4·cos(x) + (4x+5)·sen(x) ) / cos2(x)

11 EXPONENCIALES(ex)'=ex

(ef(x))'=ef(x)·f'(x)

(e10x10)' = (e10x

10)·(100x9)

(e14x4)' = (e14x

4)·(56x3)

(e3x1)' = (e3x

1)·(3x0)

(e10x5)' = (e10x

5)·(50x4)

(e13x4)' = (e13x

4)·(52x3)

(esen(x))' = (esen(x))·cos(x)(esen(3x))' = (esen(3x))·cos(3x)·3(esen(1x

2))' = (esen(1x2))·cos(1x2)·2x

(1f(x))' = (1f(x))·f'(x)·ln(1)(15x+8))' = (15x+8))·5·ln(1)(811x

7)' = (811x

7)·(77x6)·ln(8)

(95x+4))' = (95x+4))·5·ln(9)(115x

6)' = (115x

6)·(30x5)·ln(11)

(85x+11))' = (85x+11))·5·ln(8)(35x

10)' = (35x

10)·(50x9)·ln(3)

(410x+4))' = (410x+4))·10·ln(4)(410x

5)' = (410x

5)·(50x4)·ln(4)

(55x+4))' = (55x+4))·5·ln(5)(65x

11)' = (65x

11)·(55x10)·ln(6)

(28x+7))' = (28x+7))·8·ln(2)(108x

12)' = (108x

12)·(96x11)·ln(10)



(39x+11))' = (39x+11))·9·ln(3)

12 LOGARÍTMICAS

(ln(x))'=1/x(ln(f(x)))'= f'(x) / f(x)(loga(f(x)))'= f'(x) / f(x) · loga(e)

(ln(15x1))' = ( 15x0 ) / ( 15x1 ) (log11(18x9))' = ( 162x8 ) / ( 18x9 ) · log11(e) (ln(7x6))' = ( 42x5 ) / ( 7x6 ) (log5(8x7))' = ( 56x6 ) / ( 8x7 ) · log5(e) (ln(8x9))' = ( 72x8 ) / ( 8x9 ) (log4(8x9))' = ( 72x8 ) / ( 8x9 ) · log4(e) (ln(4x2))' = ( 8x1 ) / ( 4x2 ) (log10(16x8))' = ( 128x7 ) / ( 16x8 ) · log10(e) (ln(19x2))' = ( 38x1 ) / ( 19x2 ) (log11(4x2))' = ( 8x1 ) / ( 4x2 ) · log11(e)

13 DERIVACIÓN LOGARÍTMICAy=f(x)g(x)

ln(y) = ln(f(x)g(x))( ln(y) )' = ( ln(f(x)g(x)) )'( ln(y) )' = ( g(x)·ln(f(x) )'y' / y = g'(x)·ln(f(x)) + g(x)·f'(x)/f(x)y'= y·( g'(x)·ln(f(x)) + g(x)·f'(x)/f(x) ) = f(x)g(x) · ( g'(x)·ln(f(x)) + g(x)·f'(x)/f(x) )y' = ( x9x+14 )'= y' = y · ( (9x+14)·ln(x) )' = y · ( 9·ln(x) + (9x+14)·1/x ) = x9x+14 · ( 9·ln(x) + (9x+14)·1/x )La expresión se puede simplificar:y' = x9x+13·( 9·ln(x) + (9x+14) )/x

y' = ( x11x+15 )'= y' = y · ( (11x+15)·ln(x) )' = y · ( 11·ln(x) + (11x+15)·1/x ) = x11x+15 · ( 11·ln(x) + (11x+15)·1/x )La expresión se puede simplificar:y' = x11x+14·( 11·ln(x) + (11x+15) )/x



y' = ( x12x+11 )'=



y' = y · ( (12x+11)·ln(x) )' = y · ( 12·ln(x) + (12x+11)·1/x ) = x12x+11 · ( 12·ln(x) + (12x+11)·1/x )La expresión se puede simplificar:y' = x12x+10·( 12·ln(x) + (12x+11) )/x


14 RECTA TANGENTE A UNA FUNCIÓN EN UN PUNTOCalcula la ecuación de la recta tangente a f(x) = 5x2 10x en x=4La recta pasará por (4,f(4)) = (4,120)Y como pendiente m = f '(4) Como f '(x) = 10x10f '(4) = 50 y la ecuación, (y y0) = m·(xx0) será (y 120) = 50·(x4)Y la explícita y = 50·x + 80

Calcula la ecuación de la recta tangente a f(x) = 1x2 2x en x=3La recta pasará por (3,f(3)) = (3,3)Y como pendiente m = f '(3) Como f '(x) = 2x2f '(3) = 4 y la ecuación, (y y0) = m·(xx0) será (y 3) = 4·(x3)Y la explícita y = 4·x + 9




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