How do you find the derivative using limits of #f(x)=x^3-12x#?

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To find the derivative of ( f(x) = x^3 - 12x ), we use the definition of the derivative:

[ f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h} ]

First, we substitute ( f(x) ) into the formula:

[ f'(x) = \lim_{{h \to 0}} \frac{{(x + h)^3 - 12(x + h) - (x^3 - 12x)}}{h} ]

Next, we expand ( (x + h)^3 ):

[ (x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 ]

Substitute this back into the formula:

[ f'(x) = \lim_{{h \to 0}} \frac{{x^3 + 3x^2h + 3xh^2 + h^3 - 12x - 12h - x^3 + 12x}}{h} ]

Simplify the expression:

[ f'(x) = \lim_{{h \to 0}} \frac{{3x^2h + 3xh^2 + h^3 - 12h}}{h} ]

Factor out an ( h ) from the numerator:

[ f'(x) = \lim_{{h \to 0}} \frac{{h(3x^2 + 3xh + h^2 - 12)}}{h} ]

Cancel out the ( h ) in the numerator and denominator:

[ f'(x) = \lim_{{h \to 0}} (3x^2 + 3xh + h^2 - 12) ]

Now, substitute ( h = 0 ) into the expression:

[ f'(x) = 3x^2 + 0 + 0 - 12 ]

Simplify to get the derivative:

[ f'(x) = 3x^2 - 12 ]