To find the derivative \( f'(x) \) and evaluate it at \( x = 5 \), we will first compute the derivative of the function:

\[
f(x) = \frac{1}{2}x^2 + x^3 - 2x.
\]

### Step 1: Differentiate \( f(x) \)

Using the rules of differentiation:

1. The derivative of \( \frac{1}{2}x^2 \) is \( \frac{1}{2} \cdot 2x = x \).
2. The derivative of \( x^3 \) is \( 3x^2 \).
3. The derivative of \( -2x \) is \( -2 \).

Thus, the derivative \( f'(x) \) is:

\[
f'(x) = x + 3x^2 - 2.
\]

### Step 2: Evaluate \( f'(5) \)

Now, substitute \( x = 5 \) into \( f'(x) \):

\[
f'(5) = 5 + 3 \cdot 5^2 - 2.
\]

Calculate the value:

\[
f'(5) = 5 + 3 \cdot 25 - 2 = 5 + 75 - 2 = 78.
\]

### Final Answer

\[
f'(5) = 78.
\]

Would you like further details or explanations? Here are some related questions to explore:

1. What is the second derivative, \( f''(x) \), of the function?
2. How do you find the critical points of \( f(x) \)?
3. Can we determine the concavity of \( f(x) \) using the second derivative?
4. What is the value of \( f(x) \) at \( x = 5 \)?
5. How would you find the local minima and maxima of \( f(x) \)?

**Tip:** When finding derivatives, always remember the power rule: if \( f(x) = ax^n \), then \( f'(x) = n \cdot ax^{n-1} \).

Learn how to calculate the derivative f'(5) for the function f(x) = 1/2 x^2 + x^3 - 2x. Detailed step-by-step solution using differentiation rules, including the power rule. Suitable for high school students.

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