Question 506991


*[Tex \LARGE  f(x)=3x^3 -6x ]


*[Tex \LARGE  f(x+h)=3(x+h)^3 -6(x+h) ]

 
*[Tex \LARGE  f(x+h)=3(x+h)(x+h)(x+h) -6(x+h) ]


*[Tex \LARGE  f(x+h)=3(x+h)(x^2+2xh+h^2) -6(x+h) ]


*[Tex \LARGE  f(x+h)=3\left(x(x^2+2xh+h^2)+h(x^2+2xh+h^2)\right) -6(x+h) ]


*[Tex \LARGE  f(x+h)=3\left(x^3+2x^2h+xh^2+hx^2+2xh^2+h^3\right) -6(x+h) ]


*[Tex \LARGE  f(x+h)=3x^3+6x^2h+3xh^2+3hx^2+6xh^2+3h^3 -6x-6h ]


*[Tex \LARGE  f(x+h)=3x^3+9x^2h+9xh^2+3h^3 -6x-6h ]

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*[Tex \LARGE  \frac{f(x+h)-f(x)}{h} ]


*[Tex \LARGE  \frac{(3x^3+9x^2h+9xh^2+3h^3 -6x-6h)-(3x^3 -6x)}{h} ]


*[Tex \LARGE  \frac{3x^3+9x^2h+9xh^2+3h^3 -6x-6h-3x^3+6x}{h} ]


*[Tex \LARGE  \frac{9x^2h+9xh^2+3h^3-6h}{h} ]


*[Tex \LARGE  \frac{h(9x^2+9xh+3h^2-6)}{h} ]


*[Tex \LARGE  9x^2+9xh+3h^2-6 ]


*[Tex \LARGE  9x^2-6+9xh+3h^2 ]


So *[Tex \LARGE  \frac{f(x+h)-f(x)}{h} = 9x^2-6+9xh+3h^2 ]


Let me know if you need more help or if you need me to explain a step in more detail.
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Thanks,


Jim