To find $f'(2)$, we first need to determine the derivative of the given function $f(x)$.

The function is given as:

$f(x) = -\frac{3}{2}x - \frac{5}{6}x^2$

Step 1: Differentiate $f(x)$

We will differentiate term by term.

• The derivative of $-\frac{3}{2}x$ is simply $-\frac{3}{2}$.
• The derivative of $-\frac{5}{6}x^2$ (using the power rule, where $\frac{d}{dx}x^n = nx^{n-1}$) is:

$\frac{d}{dx} \left( -\frac{5}{6}x^2 \right) = -\frac{5}{6} \times 2x = -\frac{5}{3}x$

So, the derivative $f'(x)$ is:

$f'(x) = -\frac{3}{2} - \frac{5}{3}x$

Step 2: Find $f'(2)$

Now, substitute $x = 2$ into the derivative:

$f'(2) = -\frac{3}{2} - \frac{5}{3}(2)$

Simplify the expression:

$f'(2) = -\frac{3}{2} - \frac{10}{3}$

Step 3: Combine the fractions

To combine $-\frac{3}{2}$ and $-\frac{10}{3}$, we need a common denominator. The least common denominator between 2 and 3 is 6.

Convert the fractions:

$-\frac{3}{2} = -\frac{9}{6}, \quad -\frac{10}{3} = -\frac{20}{6}$

Now add the two fractions:

$f'(2) = -\frac{9}{6} - \frac{20}{6} = -\frac{29}{6}$

Final Answer:

$f'(2) = -\frac{29}{6}$

This is the value of $f'(2)$ expressed as a single fraction in simplest form.

Would you like further details on the derivative process or the simplification steps?

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Here are five related questions you might find useful:

1. How do you find the derivative of a function with multiple terms?
2. What is the power rule for differentiation?
3. Can you differentiate a polynomial of higher degree?
4. How do you find critical points using derivatives?
5. What is the significance of a derivative at a specific point?

Tip: When combining fractions with different denominators, always find the least common denominator to simplify calculations efficiently.