Derivadas

Santiago Moll Lpez

Objetivos:

IAplicar la regla de la cadena al clculo de

derivadas

IRepasar las derivas de algunas funciones

trigonomtricas

Requisitos:

IConocer las derivadas de las funciones

elementales

Objetivos:

IAplicar la regla de la cadena al clculo de

derivadas

IRepasar las derivas de algunas funciones

trigonomtricas

Requisitos:

IConocer las derivadas de las funciones

elementales

Regla de la Cadena:

I

f (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

I

f (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

I

f (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

Regla de la Cadena:

If (x) = sin(g(x))

f (x) = cos(g(x)) g (x)

I

f (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

I

f (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

Regla de la Cadena:

If (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

I

f (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

I

f (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

Regla de la Cadena:

If (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

If (x) = cos(g(x))

f (x) =sin(g(x)) g (x)

I

f (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

Regla de la Cadena:

If (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

If (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

I

f (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

Regla de la Cadena:

If (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

If (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

If (x) = tan(g(x))

f (x) = (1+ tan2(g(x))) g (x)

Regla de la Cadena:

If (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

If (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

If (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

Ejemplos:

f (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

I

f (x) = sin(x2+3x+8) f (x) = cos(x2+3x+8) (2x+3)

I

f (x) = sin(ex + x3) f (x) = cos(ex + x3) (ex +3x2)

I

f (x) = sin(ln(x)) f (x) = cos(ln(x)) 1x

Ejemplos:

f (x) = sin(g(x))

f (x) = cos(g(x)) g (x)

I

f (x) = sin(x2+3x+8) f (x) = cos(x2+3x+8) (2x+3)

I

f (x) = sin(ex + x3) f (x) = cos(ex + x3) (ex +3x2)

I

f (x) = sin(ln(x)) f (x) = cos(ln(x)) 1x

Ejemplos:

f (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

I

f (x) = sin(x2+3x+8) f (x) = cos(x2+3x+8) (2x+3)

I

f (x) = sin(ex + x3) f (x) = cos(ex + x3) (ex +3x2)

I

f (x) = sin(ln(x)) f (x) = cos(ln(x)) 1x

Ejemplos:

f (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

If (x) = sin(x2+3x+8)

f (x) = cos(x2+3x+8) (2x+3)

I

f (x) = sin(ex + x3) f (x) = cos(ex + x3) (ex +3x2)

I

f (x) = sin(ln(x)) f (x) = cos(ln(x)) 1x

Ejemplos:

f (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

If (x) = sin(x2+3x+8) f (x) = cos(x2+3x+8) (2x+3)

I

f (x) = sin(ex + x3) f (x) = cos(ex + x3) (ex +3x2)

I

f (x) = sin(ln(x)) f (x) = cos(ln(x)) 1x

Ejemplos:

f (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

If (x) = sin(x2+3x+8) f (x) = cos(x2+3x+8) (2x+3)

If (x) = sin(ex + x3)

f (x) = cos(ex + x3) (ex +3x2)

I

f (x) = sin(ln(x)) f (x) = cos(ln(x)) 1x

Ejemplos:

f (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

If (x) = sin(x2+3x+8) f (x) = cos(x2+3x+8) (2x+3)

If (x) = sin(ex + x3) f (x) = cos(ex + x3) (ex +3x2)

I

f (x) = sin(ln(x)) f (x) = cos(ln(x)) 1x

Ejemplos:

f (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

If (x) = sin(x2+3x+8) f (x) = cos(x2+3x+8) (2x+3)

If (x) = sin(ex + x3) f (x) = cos(ex + x3) (ex +3x2)

If (x) = sin(ln(x))

f (x) = cos(ln(x)) 1x

Ejemplos:

f (x) = sin(g(x)) f (x) = cos(g(x)) g (x)

If (x) = sin(x2+3x+8) f (x) = cos(x2+3x+8) (2x+3)

If (x) = sin(ex + x3) f (x) = cos(ex + x3) (ex +3x2)

If (x) = sin(ln(x)) f (x) = cos(ln(x)) 1x

Ejemplos:

f (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

I

f (x) = cos(x3+3x2) f (x) =sin(x3+3x2) (3x2+6x)

I

f (x) = cos(3ex + x) f (x) =sin(3ex + x) (3ex +1)

I

f (x) = cos(3 ln(x)) f (x) =sin(3 ln(x)) 3x

Ejemplos:

f (x) = cos(g(x))

f (x) =sin(g(x)) g (x)

I

f (x) = cos(x3+3x2) f (x) =sin(x3+3x2) (3x2+6x)

I

f (x) = cos(3ex + x) f (x) =sin(3ex + x) (3ex +1)

I

f (x) = cos(3 ln(x)) f (x) =sin(3 ln(x)) 3x

Ejemplos:

f (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

I

f (x) = cos(x3+3x2) f (x) =sin(x3+3x2) (3x2+6x)

I

f (x) = cos(3ex + x) f (x) =sin(3ex + x) (3ex +1)

I

f (x) = cos(3 ln(x)) f (x) =sin(3 ln(x)) 3x

Ejemplos:

f (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

If (x) = cos(x3+3x2)

f (x) =sin(x3+3x2) (3x2+6x)

I

f (x) = cos(3ex + x) f (x) =sin(3ex + x) (3ex +1)

I

f (x) = cos(3 ln(x)) f (x) =sin(3 ln(x)) 3x

Ejemplos:

f (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

If (x) = cos(x3+3x2) f (x) =sin(x3+3x2) (3x2+6x)

I

f (x) = cos(3ex + x) f (x) =sin(3ex + x) (3ex +1)

I

f (x) = cos(3 ln(x)) f (x) =sin(3 ln(x)) 3x

Ejemplos:

f (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

If (x) = cos(x3+3x2) f (x) =sin(x3+3x2) (3x2+6x)

If (x) = cos(3ex + x)

f (x) =sin(3ex + x) (3ex +1)

I

f (x) = cos(3 ln(x)) f (x) =sin(3 ln(x)) 3x

Ejemplos:

f (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

If (x) = cos(x3+3x2) f (x) =sin(x3+3x2) (3x2+6x)

If (x) = cos(3ex + x) f (x) =sin(3ex + x) (3ex +1)

I

f (x) = cos(3 ln(x)) f (x) =sin(3 ln(x)) 3x

Ejemplos:

f (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

If (x) = cos(x3+3x2) f (x) =sin(x3+3x2) (3x2+6x)

If (x) = cos(3ex + x) f (x) =sin(3ex + x) (3ex +1)

If (x) = cos(3 ln(x))

f (x) =sin(3 ln(x)) 3x

Ejemplos:

f (x) = cos(g(x)) f (x) =sin(g(x)) g (x)

If (x) = cos(x3+3x2) f (x) =sin(x3+3x2) (3x2+6x)

If (x) = cos(3ex + x) f (x) =sin(3ex + x) (3ex +1)

If (x) = cos(3 ln(x)) f (x) =sin(3 ln(x)) 3x

Ejemplos:

f (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

I

f (x) = tan(5x4) f (x) = (1+ tan2 (5x4)) (20x3)

I

f (x) = tan(x) f (x) = (1+ tan2 (x)) ( 12

x

)

I

f (x) = tan(cos(x)) f (x) = (1+ tan2 (cos(x))) (sin(x))

Ejemplos:

f (x) = tan(g(x))

f (x) = (1+ tan2(g(x))) g (x)

I

f (x) = tan(5x4) f (x) = (1+ tan2 (5x4)) (20x3)

I

f (x) = tan(x) f (x) = (1+ tan2 (x)) ( 12

x

)

I

f (x) = tan(cos(x)) f (x) = (1+ tan2 (cos(x))) (sin(x))

Ejemplos:

f (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

I

f (x) = tan(5x4) f (x) = (1+ tan2 (5x4)) (20x3)

I

f (x) = tan(x) f (x) = (1+ tan2 (x)) ( 12

x

)

I

f (x) = tan(cos(x)) f (x) = (1+ tan2 (cos(x))) (sin(x))

Ejemplos:

f (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

If (x) = tan(5x4)

f (x) = (1+ tan2 (5x4)) (20x3)

I

f (x) = tan(x) f (x) = (1+ tan2 (x)) ( 12

x

)

I

f (x) = tan(cos(x)) f (x) = (1+ tan2 (cos(x))) (sin(x))

Ejemplos:

f (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

If (x) = tan(5x4) f (x) = (1+ tan2 (5x4)) (20x3)

I

f (x) = tan(x) f (x) = (1+ tan2 (x)) ( 12

x

)

I

f (x) = tan(cos(x)) f (x) = (1+ tan2 (cos(x))) (sin(x))

Ejemplos:

f (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

If (x) = tan(5x4) f (x) = (1+ tan2 (5x4)) (20x3)

If (x) = tan(

x)

f (x) = (1+ tan2 (x)) ( 12

x

)

I

f (x) = tan(cos(x)) f (x) = (1+ tan2 (cos(x))) (sin(x))

Ejemplos:

f (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

If (x) = tan(5x4) f (x) = (1+ tan2 (5x4)) (20x3)

If (x) = tan(

x) f (x) = (1+ tan2 (x)) ( 12

x

)

I

f (x) = tan(cos(x)) f (x) = (1+ tan2 (cos(x))) (sin(x))

Ejemplos:

f (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

If (x) = tan(5x4) f (x) = (1+ tan2 (5x4)) (20x3)

If (x) = tan(

x) f (x) = (1+ tan2 (x)) ( 12

x

)

If (x) = tan(cos(x))

f (x) = (1+ tan2 (cos(x))) (sin(x))

Ejemplos:

f (x) = tan(g(x)) f (x) = (1+ tan2(g(x))) g (x)

If (x) = tan(5x4) f (x) = (1+ tan2 (5x4)) (20x3)

If (x) = tan(

x) f (x) = (1+ tan2 (x)) ( 12

x

)

If (x) = tan(cos(x)) f (x) = (1+ tan2 (cos(x))) (sin(x))

Repaso:

IClculo de derivadas de funciones

trigonomtricas compuestas