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We are given the function \( f(x) = \sqrt{1 - 2x} \), and we need to find the derivative \( f'(-3) \).

### Step 1: Differentiate \( f(x) \)
The function is \( f(x) = \sqrt{1 - 2x} \). To differentiate this, we will use the chain rule. Let’s rewrite the function in a more convenient form for differentiation:
\[
f(x) = (1 - 2x)^{1/2}
\]

Now, apply the chain rule:

\[
f'(x) = \frac{1}{2}(1 - 2x)^{-1/2} \cdot (-2)
\]

Simplifying this:

\[
f'(x) = \frac{-2}{2} \cdot (1 - 2x)^{-1/2} = \frac{-1}{(1 - 2x)^{1/2}}
\]

### Step 2: Evaluate \( f'(-3) \)
Now, substitute \( x = -3 \) into the derivative:

\[
f'(-3) = \frac{-1}{\sqrt{1 - 2(-3)}} = \frac{-1}{\sqrt{1 + 6}} = \frac{-1}{\sqrt{7}}
\]

Thus, the derivative \( f'(-3) = \frac{-1}{\sqrt{7}} \).

Would you like further explanation on any part of this process?

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Here are 5 related questions to expand your understanding:
1. How does the chain rule apply to more complex functions like products or quotients?
2. What are the steps to differentiate a composition of multiple functions?
3. How does implicit differentiation work, and when should you use it?
4. How do you handle differentiating logarithmic and exponential functions?
5. What is the significance of the second derivative in the context of concavity?

**Tip:** When applying the chain rule, always differentiate the outer function first, then multiply by the derivative of the inner function.

Learn how to differentiate the function f(x) = √(1 - 2x) using the chain rule, and find f'(-3). This problem demonstrates the application of differentiation rules and is suitable for Grades 11-12 calculus students.

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