How do you use the limit definition of the derivative to find the derivative of #f(x)=3x^2#?
Verification using the power rule yields the same result.
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To find the derivative of ( f(x) = 3x^2 ) using the limit definition of the derivative, follow these steps:

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

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

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

Simplify the expression: [ f'(x) = \lim_{{h \to 0}} \frac{{3x^2 + 6xh + 3h^2  3x^2}}{h} ] [ f'(x) = \lim_{{h \to 0}} \frac{{6xh + 3h^2}}{h} ]

Factor out an ( h ) from the numerator: [ f'(x) = \lim_{{h \to 0}} \frac{{h(6x + 3h)}}{h} ]

Cancel out ( h ) in the numerator and denominator: [ f'(x) = \lim_{{h \to 0}} (6x + 3h) ]

Evaluate the limit as ( h ) approaches 0: [ f'(x) = 6x ]
So, the derivative of ( f(x) = 3x^2 ) is ( f'(x) = 6x ).
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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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