# Using the limit definition, how do you find the derivative of # f(x) = 2x^2-x#?

The function's derivative is

Distribute and simplify.

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To find the derivative of ( f(x) = 2x^2 - x ) using the limit definition of the derivative, we use the formula:

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

First, we substitute the function ( 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} ]

Next, we expand and simplify the numerator:

[ 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} 4x + 2h - 1 ]

Finally, we 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 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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