Ingresa un problema...
Precálculo Ejemplos
f(x)=-2x3+7x2+xf(x)=−2x3+7x2+x
Paso 1
Considera la fórmula del cociente diferencial.
f(x+h)-f(x)hf(x+h)−f(x)h
Paso 2
Paso 2.1
Evalúa la función en x=x+hx=x+h.
Paso 2.1.1
Reemplaza la variable xx con x+hx+h en la expresión.
f(x+h)=-2(x+h)3+7(x+h)2+x+hf(x+h)=−2(x+h)3+7(x+h)2+x+h
Paso 2.1.2
Simplifica el resultado.
Paso 2.1.2.1
Elimina los paréntesis.
f(x+h)=-2(x+h)3+7(x+h)2+x+hf(x+h)=−2(x+h)3+7(x+h)2+x+h
Paso 2.1.2.2
Simplifica cada término.
Paso 2.1.2.2.1
Usa el teorema del binomio.
f(x+h)=-2(x3+3x2h+3xh2+h3)+7(x+h)2+x+hf(x+h)=−2(x3+3x2h+3xh2+h3)+7(x+h)2+x+h
Paso 2.1.2.2.2
Aplica la propiedad distributiva.
f(x+h)=-2x3-2(3x2h)-2(3xh2)-2h3+7(x+h)2+x+hf(x+h)=−2x3−2(3x2h)−2(3xh2)−2h3+7(x+h)2+x+h
Paso 2.1.2.2.3
Simplifica.
Paso 2.1.2.2.3.1
Multiplica 33 por -2−2.
f(x+h)=-2x3-6(x2h)-2(3xh2)-2h3+7(x+h)2+x+hf(x+h)=−2x3−6(x2h)−2(3xh2)−2h3+7(x+h)2+x+h
Paso 2.1.2.2.3.2
Multiplica 33 por -2−2.
f(x+h)=-2x3-6(x2h)-6(xh2)-2h3+7(x+h)2+x+hf(x+h)=−2x3−6(x2h)−6(xh2)−2h3+7(x+h)2+x+h
f(x+h)=-2x3-6(x2h)-6(xh2)-2h3+7(x+h)2+x+hf(x+h)=−2x3−6(x2h)−6(xh2)−2h3+7(x+h)2+x+h
Paso 2.1.2.2.4
Elimina los paréntesis.
f(x+h)=-2x3-6x2h-6xh2-2h3+7(x+h)2+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7(x+h)2+x+h
Paso 2.1.2.2.5
Reescribe (x+h)2(x+h)2 como (x+h)(x+h)(x+h)(x+h).
f(x+h)=-2x3-6x2h-6xh2-2h3+7((x+h)(x+h))+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7((x+h)(x+h))+x+h
Paso 2.1.2.2.6
Expande (x+h)(x+h)(x+h)(x+h) con el método PEIU (primero, exterior, interior, ultimo).
Paso 2.1.2.2.6.1
Aplica la propiedad distributiva.
f(x+h)=-2x3-6x2h-6xh2-2h3+7(x(x+h)+h(x+h))+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7(x(x+h)+h(x+h))+x+h
Paso 2.1.2.2.6.2
Aplica la propiedad distributiva.
f(x+h)=-2x3-6x2h-6xh2-2h3+7(x⋅x+xh+h(x+h))+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7(x⋅x+xh+h(x+h))+x+h
Paso 2.1.2.2.6.3
Aplica la propiedad distributiva.
f(x+h)=-2x3-6x2h-6xh2-2h3+7(x⋅x+xh+hx+h⋅h)+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7(x⋅x+xh+hx+h⋅h)+x+h
f(x+h)=-2x3-6x2h-6xh2-2h3+7(x⋅x+xh+hx+h⋅h)+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7(x⋅x+xh+hx+h⋅h)+x+h
Paso 2.1.2.2.7
Simplifica y combina los términos similares.
Paso 2.1.2.2.7.1
Simplifica cada término.
Paso 2.1.2.2.7.1.1
Multiplica xx por xx.
f(x+h)=-2x3-6x2h-6xh2-2h3+7(x2+xh+hx+h⋅h)+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7(x2+xh+hx+h⋅h)+x+h
Paso 2.1.2.2.7.1.2
Multiplica hh por hh.
f(x+h)=-2x3-6x2h-6xh2-2h3+7(x2+xh+hx+h2)+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7(x2+xh+hx+h2)+x+h
f(x+h)=-2x3-6x2h-6xh2-2h3+7(x2+xh+hx+h2)+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7(x2+xh+hx+h2)+x+h
Paso 2.1.2.2.7.2
Suma xhxh y hxhx.
Paso 2.1.2.2.7.2.1
Reordena xx y hh.
f(x+h)=-2x3-6x2h-6xh2-2h3+7(x2+hx+hx+h2)+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7(x2+hx+hx+h2)+x+h
Paso 2.1.2.2.7.2.2
Suma hxhx y hxhx.
f(x+h)=-2x3-6x2h-6xh2-2h3+7(x2+2hx+h2)+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7(x2+2hx+h2)+x+h
f(x+h)=-2x3-6x2h-6xh2-2h3+7(x2+2hx+h2)+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7(x2+2hx+h2)+x+h
f(x+h)=-2x3-6x2h-6xh2-2h3+7(x2+2hx+h2)+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7(x2+2hx+h2)+x+h
Paso 2.1.2.2.8
Aplica la propiedad distributiva.
f(x+h)=-2x3-6x2h-6xh2-2h3+7x2+7(2hx)+7h2+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7x2+7(2hx)+7h2+x+h
Paso 2.1.2.2.9
Multiplica 22 por 77.
f(x+h)=-2x3-6x2h-6xh2-2h3+7x2+14hx+7h2+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7x2+14hx+7h2+x+h
f(x+h)=-2x3-6x2h-6xh2-2h3+7x2+14hx+7h2+x+hf(x+h)=−2x3−6x2h−6xh2−2h3+7x2+14hx+7h2+x+h
Paso 2.1.2.3
La respuesta final es -2x3-6x2h-6xh2-2h3+7x2+14hx+7h2+x+h−2x3−6x2h−6xh2−2h3+7x2+14hx+7h2+x+h.
-2x3-6x2h-6xh2-2h3+7x2+14hx+7h2+x+h−2x3−6x2h−6xh2−2h3+7x2+14hx+7h2+x+h
-2x3-6x2h-6xh2-2h3+7x2+14hx+7h2+x+h−2x3−6x2h−6xh2−2h3+7x2+14hx+7h2+x+h
-2x3-6x2h-6xh2-2h3+7x2+14hx+7h2+x+h−2x3−6x2h−6xh2−2h3+7x2+14hx+7h2+x+h
Paso 2.2
Reordena.
Paso 2.2.1
Mueve x2x2.
-2x3-6hx2-6xh2-2h3+7x2+14hx+7h2+x+h−2x3−6hx2−6xh2−2h3+7x2+14hx+7h2+x+h
Paso 2.2.2
Mueve xx.
-2x3-6hx2-6h2x-2h3+7x2+14hx+7h2+x+h−2x3−6hx2−6h2x−2h3+7x2+14hx+7h2+x+h
Paso 2.2.3
Mueve xx.
-2x3-6hx2-6h2x-2h3+7x2+14hx+7h2+h+x−2x3−6hx2−6h2x−2h3+7x2+14hx+7h2+h+x
Paso 2.2.4
Mueve 7x27x2.
-2x3-6hx2-6h2x-2h3+14hx+7h2+7x2+h+x−2x3−6hx2−6h2x−2h3+14hx+7h2+7x2+h+x
Paso 2.2.5
Mueve 14hx14hx.
-2x3-6hx2-6h2x-2h3+7h2+14hx+7x2+h+x−2x3−6hx2−6h2x−2h3+7h2+14hx+7x2+h+x
Paso 2.2.6
Mueve -2x3−2x3.
-6hx2-6h2x-2h3-2x3+7h2+14hx+7x2+h+x−6hx2−6h2x−2h3−2x3+7h2+14hx+7x2+h+x
Paso 2.2.7
Mueve -6hx2−6hx2.
-6h2x-2h3-6hx2-2x3+7h2+14hx+7x2+h+x−6h2x−2h3−6hx2−2x3+7h2+14hx+7x2+h+x
Paso 2.2.8
Reordena -6h2x−6h2x y -2h3−2h3.
-2h3-6h2x-6hx2-2x3+7h2+14hx+7x2+h+x−2h3−6h2x−6hx2−2x3+7h2+14hx+7x2+h+x
-2h3-6h2x-6hx2-2x3+7h2+14hx+7x2+h+x−2h3−6h2x−6hx2−2x3+7h2+14hx+7x2+h+x
Paso 2.3
Obtén los componentes de la definición.
f(x+h)=-2h3-6h2x-6hx2-2x3+7h2+14hx+7x2+h+xf(x+h)=−2h3−6h2x−6hx2−2x3+7h2+14hx+7x2+h+x
f(x)=-2x3+7x2+xf(x)=−2x3+7x2+x
f(x+h)=-2h3-6h2x-6hx2-2x3+7h2+14hx+7x2+h+xf(x+h)=−2h3−6h2x−6hx2−2x3+7h2+14hx+7x2+h+x
f(x)=-2x3+7x2+xf(x)=−2x3+7x2+x
Paso 3
Inserta los componentes.
f(x+h)-f(x)h=-2h3-6h2x-6hx2-2x3+7h2+14hx+7x2+h+x-(-2x3+7x2+x)hf(x+h)−f(x)h=−2h3−6h2x−6hx2−2x3+7h2+14hx+7x2+h+x−(−2x3+7x2+x)h
Paso 4
Paso 4.1
Simplifica el numerador.
Paso 4.1.1
Aplica la propiedad distributiva.
-2h3-6h2x-6hx2-2x3+7h2+14hx+7x2+h+x-(-2x3)-(7x2)-xh−2h3−6h2x−6hx2−2x3+7h2+14hx+7x2+h+x−(−2x3)−(7x2)−xh
Paso 4.1.2
Simplifica.
Paso 4.1.2.1
Multiplica -2−2 por -1−1.
-2h3-6h2x-6hx2-2x3+7h2+14hx+7x2+h+x+2x3-(7x2)-xh−2h3−6h2x−6hx2−2x3+7h2+14hx+7x2+h+x+2x3−(7x2)−xh
Paso 4.1.2.2
Multiplica 77 por -1−1.
-2h3-6h2x-6hx2-2x3+7h2+14hx+7x2+h+x+2x3-7x2-xh−2h3−6h2x−6hx2−2x3+7h2+14hx+7x2+h+x+2x3−7x2−xh
-2h3-6h2x-6hx2-2x3+7h2+14hx+7x2+h+x+2x3-7x2-xh−2h3−6h2x−6hx2−2x3+7h2+14hx+7x2+h+x+2x3−7x2−xh
Paso 4.1.3
Suma -2x3−2x3 y 2x32x3.
-2h3-6h2x-6hx2+7h2+14hx+7x2+h+x+0-7x2-xh−2h3−6h2x−6hx2+7h2+14hx+7x2+h+x+0−7x2−xh
Paso 4.1.4
Suma -2h3−2h3 y 00.
-2h3-6h2x-6hx2+7h2+14hx+7x2+h+x-7x2-xh−2h3−6h2x−6hx2+7h2+14hx+7x2+h+x−7x2−xh
Paso 4.1.5
Resta 7x27x2 de 7x27x2.
-2h3-6h2x-6hx2+7h2+14hx+h+x+0-xh−2h3−6h2x−6hx2+7h2+14hx+h+x+0−xh
Paso 4.1.6
Suma -2h3−2h3 y 00.
-2h3-6h2x-6hx2+7h2+14hx+h+x-xh−2h3−6h2x−6hx2+7h2+14hx+h+x−xh
Paso 4.1.7
Resta xx de xx.
-2h3-6h2x-6hx2+7h2+14hx+h+0h−2h3−6h2x−6hx2+7h2+14hx+h+0h
Paso 4.1.8
Suma -2h3-6h2x-6hx2+7h2+14hx+h−2h3−6h2x−6hx2+7h2+14hx+h y 00.
-2h3-6h2x-6hx2+7h2+14hx+hh−2h3−6h2x−6hx2+7h2+14hx+hh
Paso 4.1.9
Factoriza hh de -2h3-6h2x-6hx2+7h2+14hx+h−2h3−6h2x−6hx2+7h2+14hx+h.
Paso 4.1.9.1
Factoriza hh de -2h3−2h3.
h(-2h2)-6h2x-6hx2+7h2+14hx+hhh(−2h2)−6h2x−6hx2+7h2+14hx+hh
Paso 4.1.9.2
Factoriza hh de -6h2x−6h2x.
h(-2h2)+h(-6hx)-6hx2+7h2+14hx+hhh(−2h2)+h(−6hx)−6hx2+7h2+14hx+hh
Paso 4.1.9.3
Factoriza hh de -6hx2−6hx2.
h(-2h2)+h(-6hx)+h(-6x2)+7h2+14hx+hhh(−2h2)+h(−6hx)+h(−6x2)+7h2+14hx+hh
Paso 4.1.9.4
Factoriza hh de 7h27h2.
h(-2h2)+h(-6hx)+h(-6x2)+h(7h)+14hx+hhh(−2h2)+h(−6hx)+h(−6x2)+h(7h)+14hx+hh
Paso 4.1.9.5
Factoriza hh de 14hx14hx.
h(-2h2)+h(-6hx)+h(-6x2)+h(7h)+h(14x)+hhh(−2h2)+h(−6hx)+h(−6x2)+h(7h)+h(14x)+hh
Paso 4.1.9.6
Eleva hh a la potencia de 11.
h(-2h2)+h(-6hx)+h(-6x2)+h(7h)+h(14x)+h1hh(−2h2)+h(−6hx)+h(−6x2)+h(7h)+h(14x)+h1h
Paso 4.1.9.7
Factoriza hh de h1h1.
h(-2h2)+h(-6hx)+h(-6x2)+h(7h)+h(14x)+h⋅1hh(−2h2)+h(−6hx)+h(−6x2)+h(7h)+h(14x)+h⋅1h
Paso 4.1.9.8
Factoriza hh de h(-2h2)+h(-6hx)h(−2h2)+h(−6hx).
h(-2h2-6hx)+h(-6x2)+h(7h)+h(14x)+h⋅1hh(−2h2−6hx)+h(−6x2)+h(7h)+h(14x)+h⋅1h
Paso 4.1.9.9
Factoriza h de h(-2h2-6hx)+h(-6x2).
h(-2h2-6hx-6x2)+h(7h)+h(14x)+h⋅1h
Paso 4.1.9.10
Factoriza h de h(-2h2-6hx-6x2)+h(7h).
h(-2h2-6hx-6x2+7h)+h(14x)+h⋅1h
Paso 4.1.9.11
Factoriza h de h(-2h2-6hx-6x2+7h)+h(14x).
h(-2h2-6hx-6x2+7h+14x)+h⋅1h
Paso 4.1.9.12
Factoriza h de h(-2h2-6hx-6x2+7h+14x)+h⋅1.
h(-2h2-6hx-6x2+7h+14x+1)h
h(-2h2-6hx-6x2+7h+14x+1)h
h(-2h2-6hx-6x2+7h+14x+1)h
Paso 4.2
Reduce la expresión mediante la cancelación de los factores comunes.
Paso 4.2.1
Cancela el factor común de h.
Paso 4.2.1.1
Cancela el factor común.
h(-2h2-6hx-6x2+7h+14x+1)h
Paso 4.2.1.2
Divide -2h2-6hx-6x2+7h+14x+1 por 1.
-2h2-6hx-6x2+7h+14x+1
-2h2-6hx-6x2+7h+14x+1
Paso 4.2.2
Simplifica la expresión.
Paso 4.2.2.1
Mueve h.
-2h2-6xh-6x2+7h+14x+1
Paso 4.2.2.2
Mueve 7h.
-2h2-6xh-6x2+14x+7h+1
Paso 4.2.2.3
Mueve -2h2.
-6xh-6x2-2h2+14x+7h+1
Paso 4.2.2.4
Reordena -6xh y -6x2.
-6x2-6xh-2h2+14x+7h+1
-6x2-6xh-2h2+14x+7h+1
-6x2-6xh-2h2+14x+7h+1
-6x2-6xh-2h2+14x+7h+1
Paso 5