The function given is $f(x) = 2x^2 - x$. To find the derivative using the definition of the derivative, we use the following limit definition:

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

Step-by-step process:

1. Compute $f(x+h)$: Substitute $x+h$ into the function $f(x)$: $f(x+h) = 2(x+h)^2 - (x+h)$ Expanding this: $f(x+h) = 2(x^2 + 2xh + h^2) - (x+h) = 2x^2 + 4xh + 2h^2 - x - h$

2. Set up the difference $f(x+h) - f(x)$: $f(x+h) - f(x) = \left( 2x^2 + 4xh + 2h^2 - x - h \right) - (2x^2 - x)$ Simplify the expression: $f(x+h) - f(x) = 4xh + 2h^2 - h$

3. Divide by $h$: $\frac{f(x+h) - f(x)}{h} = \frac{4xh + 2h^2 - h}{h} = 4x + 2h - 1$

4. Take the limit as $h \to 0$: $\lim_{h \to 0} (4x + 2h - 1) = 4x - 1$

Thus, the derivative of the function is:

$f'(x) = 4x - 1$

Let me know if you'd like further clarification on any step!

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Here are 5 related questions:

1. What is the second derivative of $f(x) = 2x^2 - x$?
2. How can you apply this derivative to find the slope of the tangent line at any point on the curve?
3. What is the derivative of $f(x) = x^3$ using the same method?
4. How can you interpret the result of $f'(x) = 4x - 1$ geometrically?
5. Can you compute the derivative of a function involving trigonometric terms using the definition of the derivative?

Tip: When expanding binomials like $(x+h)^2$, use the formula $(x+h)^2 = x^2 + 2xh + h^2$ to simplify the expression quickly.