Let $f(x) = -3 x^3 + 9 x + 4$ , how do you use the limit definition of the derivative to calculate the derivative of f?

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Answer 1 · 2015-04-15T15:52:55

The definition is:

$f'(x)=lim_(hrarr0)(f(x+h)-f(x))/h$ .

So:

$lim_(hrarr0)(-3(x+h)^3+9(x+h)+4-(-3x^3+9x+4))/h=$

$=lim_(hrarr0)(-3(x^3+3x^2h+3xh^2+h^3)+9x+9h+4+3x^3-9x-4)/h=$

$=lim_(hrarr0)(-3x^3-9x^2h-9xh^2-3h^3+9x+9h+4+3x^3-9x-4)/h=$

$=lim_(hrarr0)(-9x^2h-9xh^2-3h^3+9h)/h=$

$=lim_(hrarr0)(-9x^2-9xh-3h^2+9)=-9x^2+9$ ,

that is the derivative of the function.

Let f(x) = -3 x^3 + 9 x + 4f(x) = -3 x^3 + 9 x + 4, how do you use the limit definition of the derivative to calculate the derivative of f?