# How do you find f'(x) using the definition of a derivative for #f(x)= 2x^2-x #?

See the explanation.

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To find ( f'(x) ) using the definition of a derivative for ( f(x) = 2x^2 - x ), follow these steps:

- Recall the definition of the derivative: ( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ).
- Substitute the given function into the definition: ( 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 + 2hx + h^2) - x - h - 2x^2 + x}{h} ).
- Combine like terms: ( f'(x) = \lim_{h \to 0} \frac{2x^2 + 4hx + 2h^2 - x - h - 2x^2 + x}{h} ).
- Further simplify: ( f'(x) = \lim_{h \to 0} \frac{4hx + 2h^2 - h}{h} ).
- Factor out ( h ) from the numerator: ( f'(x) = \lim_{h \to 0} \frac{h(4x + 2h - 1)}{h} ).
- Cancel out ( h ) from the numerator and denominator: ( f'(x) = \lim_{h \to 0} (4x + 2h - 1) ).
- Evaluate the limit as ( h ) approaches 0: ( f'(x) = 4x - 1 ).

Therefore, ( f'(x) = 4x - 1 ) is the derivative of ( f(x) = 2x^2 - x ) using the definition of a derivative.

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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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