How do you use the limit definition of the derivative to find the derivative of #f(x)=3x^2+x+5#?
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To find the derivative of ( f(x) = 3x^2 + x + 5 ) using the limit definition of the derivative, follow these steps:

Write down 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 + x + 5 ) into the limit definition: [ f'(x) = \lim_{h \to 0} \frac{[3(x + h)^2 + (x + h) + 5]  [3x^2 + x + 5]}{h} ]

Expand and simplify the expression inside the limit: [ f'(x) = \lim_{h \to 0} \frac{3(x^2 + 2xh + h^2) + x + h + 5 + 3x^2  x  5}{h} ] [ f'(x) = \lim_{h \to 0} \frac{3x^2  6xh  3h^2 + x + h + 5 + 3x^2  x  5}{h} ] [ f'(x) = \lim_{h \to 0} \frac{6xh  3h^2 + h}{h} ]

Factor out ( h ) and cancel: [ f'(x) = \lim_{h \to 0} \frac{h(6x  3h + 1)}{h} ] [ f'(x) = \lim_{h \to 0} (6x  3h + 1) ]

Evaluate the limit as ( h ) approaches 0: [ f'(x) = 6x + 1 ]
Therefore, the derivative of ( f(x) = 3x^2 + x + 5 ) is ( f'(x) = 6x + 1 ).
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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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