Given the function f, of, x, equals, minus, start fraction, 3, divided by, 2, x, end fraction, minus, start fraction, 5, divided by, 6, x, squared, end fraction, comma*f*(*x*)=−2*x*3​−6*x*25​, find f, prime, of, 2, .*f*′(2). Express your answer as a single fraction in simplest form.

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.

Given the function f(x) = -3/2x - 5/6x^2, find f'(2). Express your answer as a single fraction in simplest form.

Learn how to find the derivative of f(x) = -3/2x - 5/6x^2 and compute f'(2). This step-by-step guide covers essential calculus concepts such as differentiation and the power rule. Suitable for students in Grades 11-12.

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