x g x f x g x g x 2 2 2 2 2 5 1 1 5 10 1 x x x x x x x 5 2 g x lim x h x h lim

# X g x f x g x g x 2 2 2 2 2 5 1 1 5 10 1 x x x x x x

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x g x f x g x g x ( ) ( ) ( ) ( ) ( ) ( ) [ ( = ' ' ' )] ( ) ( )( ) ( ) ( 2 2 2 2 2 5 1 1 5 10 1 = ⋅ − + + − + = + + x x x x x x − + x 5 2 ) g x lim x h x h lim x h x h h h ' ( ) ( ) ( ) = + + − − + = − − + = 0 0 5 5 1 f x lim x h x h lim x hx h h h ' ( ) ( ) ( ) = + + − + = + + 0 2 2 0 2 1 1 2 2 2 0 2 2 = + = x h lim x h x h ( ) g x f x ( ) ( ) f x g x ( ) ( ) 015 Aplicando la fórmula: p x f x g x f x g ' ' ' ( ) ( ) ( ) ( ) = + ( ) ( ) ( ) ( ) ( ) x lim x h x h x x l h = = + + + 0 2 2 2 4 4 1 4 im x h x h lim x hx h x h h h 0 0 2 2 2 1 1 2 ( ) ( ) ( + + − + = = + + x x lim x h x h lim x h x h h + + + = = + + 1 4 2 2 0 0 ) ( ) ( ) ( 1 4 1 2 1 4 3 2 4 2 2 2 ) ( ) ( ) + = + + = + x x x x x x p x f x g x x x x x x p x ( ) ( ) ( ) ( )( ) ( ) = = + = + = 2 3 2 4 1 4 4 ' lim x h x h x h x x x h h 0 3 2 3 2 4 4 4 4 ( ) ( ) ( ) ( ) + + + + + = = + + + + + + lim x hx h x h x hx h x h h 0 3 2 2 3 2 2 3 3 2 4 4 4 x x x h lim hx h x h hx h h h h 3 2 0 2 2 3 2 4 4 3 3 2 4 + + = = + + + + = + + + + = = + lim x hx h x h x x h 0 2 2 2 3 3 2 4 3 2 4 ( ) 014 Derivada de una función
437 Halla la derivada de las siguientes funciones. a) f ( x ) = 5 sen x + 3 cos x b) f ( x ) = (5 x 2 sen x ) + ( x cos x ) a) f ' ( x ) = 5 · cos x + 3 · ( sen x ) = 5 cos x 3 sen x b) f ' ( x ) = (5 · 2 x · sen x + 5 x 2 · cos x ) + (1 · cos x + x · ( sen x )) = = 10 x sen x + 5 x 2 cos x + cos x x sen x Obtén la derivada de estas funciones. a) f ( x ) = e x tg x b) f ( x ) = 3 x 2 arc s en x a) f ' ( x ) = e x · tg x + e x · (1 + tg 2 x ) = e x (1 + tg x + tg 2 x ) Halla la derivada de estas funciones aplicando la regla de la cadena. a) f ( x ) = ln ( cos x ) c) f ( x ) = ( x 4 + 2) 9 b) f ( x ) = cos (ln x ) d) f ( x ) = x Calcula la derivada de estas funciones. a) f ( x ) = sen c) f ( x ) = ln b) f ( x ) = 3 sen x 2 + 2 sen 2 x d) f ( x ) = f x e x x e x x x x ' ( ) = + ( ) = + + ( ) + ( ) 1 1 2 1 2 2 2 1 1 2 1 c) f x x x x x x ' ( ) ( ) ( ) ( ) ( ) = + + ⋅ − = 1 1 1 1 1 1 1 1 2 1 2 x 2 b) f x cos x x sen x cos x x cos x ' ( ) = + = + 3 2 2 2 6 4 2 2 sen x cos x a) f x cos x x x x x x ' ( ) ( ) ( ) ( ) = + + + = + 2 2 1 2 3 1 2 3 2 3 2 3 cos x x x x 2 2 3 2 3 + + e x + ( ) 1 2 1 1 + x x x x 2 3 + 021 d) f x x x x x x ' ( ) ( ) = + + + = = + + 1 2 1 1 2 2 1 6 2 1 3 3 3 1 2 2 3 x x x x 3 3 3 3 2 1 5 1 2 1 + = + + c) f x x x x x ' ( ) ( ) ( ) = + = + 9 2 4 36 2 4 8 3 3 4 8 b) f x sen x x ' ( ) (ln ) = − 1 a) f x cos x sen x tg x ' ( ) ( ) = ⋅ − = − 1 2 1 3 x + 020 b) f x x x ' ( ) = 6 1 1 2 019 018 10 SOLUCIONARIO
438 Halla los intervalos de crecimiento y decrecimiento de las siguientes funciones. a) f ( x ) = x 2 6 x + 5 b) f ( x ) = 8 x + x 2 a) f ' ( x ) = 2 x 6 2 x 6 = 0 x = 3 La función es decreciente en ( , 3) y es creciente en (3, + ). b) f ' ( x ) = 8 + 2 x 8 + 2 x = 0 x = − 4 La función es decreciente en ( , 4) y es creciente en ( 4, + ). Determina los intervalos de crecimiento y los máximos y mínimos de estas funciones. a) f ( x ) = x 3 3 x b) f ( x ) = 2 x a) f ' ( x ) = 3 x 2 3 3 x 2 3 = 0 x = ± 1 La función es creciente en ( , 1) (1, + ) y es decreciente en ( 1, 1). Presenta un máximo en x = − 1 y un mínimo en x = 1.

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