# How do you find the derivative of #f(x) = x^2+3x+1# using the limit definition?

Do some substituting and algebra to get

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To find the derivative of ( f(x) = x^2 + 3x + 1 ) using the limit definition, you would start by applying the definition of the derivative, which is:

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

Substitute ( f(x) = x^2 + 3x + 1 ) into the formula:

[ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 + 3(x + h) + 1 - (x^2 + 3x + 1)}{h} ]

Simplify the expression inside the limit:

[ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 + 3x + 3h + 1 - x^2 - 3x - 1}{h} ]

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

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

Now, as ( h ) approaches 0, the limit becomes:

[ f'(x) = 2x + 3 ]

So, the derivative of ( f(x) = x^2 + 3x + 1 ) is ( f'(x) = 2x + 3 ).

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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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