How do you use the limit definition of the derivative to find the derivative of #f(x)=x^3#?
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To find the derivative of ( f(x) = x^3 ) using the limit definition of the derivative, follow these steps:

Start with the limit definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x + h)  f(x)}{h} ]

Substitute ( f(x) = x^3 ) into the formula: [ f'(x) = \lim_{h \to 0} \frac{(x + h)^3  x^3}{h} ]

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

Substitute the expanded expression into the derivative formula: [ f'(x) = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3  x^3}{h} ]

Simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h} ]

Factor out ( h ) from the numerator: [ f'(x) = \lim_{h \to 0} \frac{h(3x^2 + 3xh + h^2)}{h} ]

Cancel out ( h ) from the numerator and denominator: [ f'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2) ]

Substitute ( h = 0 ) into the expression: [ f'(x) = 3x^2 ]
So, the derivative of ( f(x) = x^3 ) is ( f'(x) = 3x^2 ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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