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Derivadas e Integrales: Reto para Sexto Semestre

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Juego de cálculo diferencial e integral

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Derivadas e Integrales: Reto para Sexto Semestre
 

Derivadas e Integrales: Reto para Sexto SemestreOnline version

Juego de cálculo diferencial e integral

by Yansaret Mendez Avila
1

Deriva la función f(x)=3x^4−2x^3+5. ¿Cuál es f'(x)?

2

Si g(x)=x^2·e^x, ¿g'(x) usando producto se obtiene?

3

Deriva h(x)=sin(3x).

4

Encuentra la derivada de y=x^5/(z) si z es constante respecto a x.

5

Indicar la regla correcta para f(x)=ln(x^2+1).

6

Calcula la integral indefinida ∫(3x^2) dx.

7

Evalúa la integral definida ∫_0^2 (2x) dx.

8

Si F'(x)=2x y F(0)=5, ¿F(3)?

9

Deriva la función compuesta p(x)=cos(2x^3).

10

Determina la regla de la integral de potencia: ∫x^n dx.

Explicación

Aplicación de la regla de potencias a cada término.

Producto con la regla de derivación: (u v)'=u'v+uv'.

Cadena: la derivada de sin(kx) es k cos(kx).

La derivada respecto a x de x^n es n x^{n-1}; z constante actúa como const.

Derivada de ln(u)=u'/u; con u=x^2+1, u'=2x.

Regla de potencias: ∫x^n dx = x^{n+1}/(n+1) + C.

Resultado: x^2 evaluado 0 a 2 = 4.

Integra: F(x)=x^2+C; C=5, entonces F(3)=9+5=14; ajusto: corregir es 11? Ajustemos: con F'(x)=2x, F(x)=x^2+C; F(0)=5 ⇒ C=5; F(3)=9+5=14.

Cadena dos veces: derivada de cos u es −sin u·u' con u=2x^3 y u'=6x^2.

Regla: ∫x^n dx = x^{n+1}/(n+1) + C para n≠−1.

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