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

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To find the derivative of ( f(x) = x^3 - 12x ) using the limit process, you use the definition of the derivative:

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

First, find ( f(x + h) ):

[ f(x + h) = (x + h)^3 - 12(x + h) ]

Next, find ( f(x) ):

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

Then, substitute both into the definition of the derivative and simplify:

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

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

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

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

Now, plug in ( h = 0 ) to find the derivative:

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

So, the derivative of ( f(x) = x^3 - 12x ) is ( f'(x) = 3x^2 - 12 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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