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

Expand and simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{2(x^2 + 2xh + h^2)  x  h  2x^2 + x}{h} ] [ f'(x) = \lim_{h \to 0} \frac{2x^2 + 4xh + 2h^2  x  h  2x^2 + x}{h} ] [ f'(x) = \lim_{h \to 0} \frac{4xh + 2h^2  h}{h} ] [ f'(x) = \lim_{h \to 0} \frac{h(4x + 2h  1)}{h} ]

Cancel out the ( h ) in the numerator and denominator: [ f'(x) = \lim_{h \to 0} (4x + 2h  1) ]

Evaluate the limit as ( h ) approaches 0: [ f'(x) = 4x  1 ]
So, the derivative of ( f(x) = 2x^2  x ) is ( f'(x) = 4x  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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