h h 7 4 7 7 7 4 7 h h lim h h h 2 30 28 4 28 4 a f f lim f h f h lim h h 2 4 2

H h 7 4 7 7 7 4 7 h h lim h h h 2 30 28 4 28 4 a f f

This preview shows page 434 - 437 out of 640 pages.

h h ' ' ( ) ( ) ( ) ( ) ( 7 4 7 7 7 4 7 0 0 = = + = + + = + = h h lim h h h ) 2 30 28 4 28 4 0 a) f f lim f h f h lim h h ' ' ( ) ( ) ( ) ( ) ( 2 4 2 2 2 4 2 0 0 = = + = + + = + = h h lim h h h ) 2 10 8 4 8 4 0 f x lim f x h f x h lim x h x h h ' ( ) ( ) ( ) ( ) ( = + = + + 0 0 4 2 4 + = + = 2 4 4 4 4 0 ) h lim x h x h h 009 Derivada de una función
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435 Halla la derivada de las siguientes funciones, utilizando la definición de derivada del producto de un número por una función. a) f ( x ) = 8 x 3 b) c) f ( x ) = − 5 x 2 f ' ( x ) = 8 · 3 x 2 = 24 x 2 c) lim x h x h lim x hx h x h lim h h h 0 2 2 0 2 2 2 2 ( ) + = + + = 0 2 0 2 2 2 5 2 10 hx h h lim x h x f x x x h + = + = = − ⋅ = − ( ) ( ) ' b) lim x h x h lim x h x h x h x lim x h h h 0 0 0 1 + = + + + ( ) = + + = = = h x x f x x x 1 2 4 1 2 2 ' ( ) a) lim x h x h lim x hx h x h x h h 0 3 3 0 3 2 2 3 3 3 3 ( ) + = + + + h lim hx h x h h lim x hx h h h = + + = = + + = 0 2 2 3 0 2 2 3 3 3 3 ( ) 3 2 x f x x ( ) = 4 013 g x lim x h x h lim x h x h l h h ' ( ) ( ) ( ) = + + + = 0 2 2 0 3 3 im x h x h lim x h x h lim x h h h 0 2 2 0 0 1 1 3 3 3 ( ) + + + + = 2 2 2 2 2 0 2 2 3 2 + + = = + x hx h h x h x lim x h h x h h ( ) ( ) x x x x x 2 4 3 3 3 2 3 3 2 + = + = b) g x lim g x h g x h lim x h h h ' ( ) ( ) ( ) ( ) ( = + = + + 0 0 2 3 x h x x h lim x h x h x x h + + = = + + + ) ( ) ( ) 2 0 2 2 3 1 3 3 1 3 h lim x hx h x h x x hx h h x h = = + + + + 0 2 4 2 3 2 2 2 2 3 6 3 2 ( h x lim x hx hx x h x h x x h ) ( ) 2 2 0 4 3 2 2 2 3 6 3 2 3 = = + + + = 4 4 3 3 2 3 2 = x x x x f x lim x h x h lim x h x h l h h ' ( ) ( ) ( ) = + + + = 0 0 2 2 7 7 2 2 im x h x h lim x hx h x h l h h 0 0 2 2 2 7 7 7 2 4 2 2 7 + + + + + = + im x h x h 0 4 2 7 4 ( ) + = + a) f x lim f x h f x h lim x h x h h ' ( ) ( ) ( ) ( ) ( = + = + + 0 0 7 2 + + = = + + + + h x x h lim x h x hx h x h ) ( ) 2 2 0 2 2 7 2 7 7 2 4 2 7 = + + = = + + = 2 7 4 2 7 4 2 7 2 0 2 0 x h lim h hx h h lim x h h h ( ) + 4 x 10 SOLUCIONARIO
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436 Calcula el producto de las funciones f ( x ) = x 2 4 y g ( x ) = x + 1, y después halla su derivada. Comprueba que el resultado es el mismo que si aplicamos la fórmula de la derivada del producto de funciones. Halla las derivadas de las funciones f ( x ) = x 2 + 1 y g ( x ) = − x + 5. ¿Cuál es la derivada del cociente ? ¿Y la derivada del cociente ? Calcula la derivada de esta función, indicando los pasos que sigues para hallarla. f ( x ) = x 2 + 2 x Se aplica la derivada de la suma de funciones, la derivada de la función potencial ( n = 2), la derivada del producto de un número por una función y la derivada de la función identidad: f ' ( x ) = 2 x 2 1 + 2 1 = 2 x + 2 Halla la derivada de la siguiente función: Se aplica la derivada del cociente de funciones. Para derivar la función del numerador se usa la derivada de la suma de funciones, la derivada de la función identidad y la derivada de la función constante. Para obtener la derivada del denominador se aplica la derivada del producto de un número por una función y la derivada de la función potencial ( n = 5): f x x x x x x x x ' ( ) ( ) ( ) = = + 1 2 3 2 5 2 8 30 4 5 4 5 2 5 4 10 = + 4 15 2 6 x x f x x x ( ) = 3 2 5 017 016 g x f x g x f x g x f x f x ( ) ( ) ( ) ( ) ( ) ( ) [ ( = ' ' ' )] ( )( ) ( ) ( ) ( 2 2 2 2 2 1 1 5 2 1 10 1 = + − − + + = x x x x x x x 2 2 1 + ) f x g x f
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