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find derivative using definition of function

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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Would you like any more details or have any questions?

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.

Find the derivative using the definition of function.

Learn how to find the derivative of f(x) = 2x^2 - x using the limit definition of a derivative. This problem covers key calculus concepts, including limits and the definition of a derivative. Suitable for advanced high school students and college calculus.

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