How do you use the limit definition of the derivative to find the derivative of #f(x)=3x^2+3x+3#?
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To find the derivative of ( f(x) = 3x^2 + 3x + 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 the function ( f(x) = 3x^2 + 3x + 3 ) into the formula: [ f'(x) = \lim_{h \to 0} \frac{(3(x + h)^2 + 3(x + h) + 3)  (3x^2 + 3x + 3)}{h} ]

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

Distribute and simplify further: [ f'(x) = \lim_{h \to 0} \frac{3x^2 + 6xh + 3h^2 + 3x + 3h + 3  3x^2  3x  3}{h} ] [ f'(x) = \lim_{h \to 0} \frac{6xh + 3h^2 + 3h}{h} ]

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

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

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