https://file.aiquickdraw.com/mathbot/file/1726389887983fhb8w7t3.jpg

The problem in the image asks to find \( f'(g(x)) \) given that:

\[
\frac{d}{dx} \left[ f(x^2 - 2x) \right] = x^2 - 3x + 2
\]
and
\[
g(x) = x^2 - 2x.
\]

### Step 1: Differentiate \( f(x^2 - 2x) \)

The expression involves differentiating \( f(u) \), where \( u = x^2 - 2x \), by applying the chain rule. The chain rule states:

\[
\frac{d}{dx} \left[ f(u) \right] = f'(u) \cdot \frac{du}{dx}.
\]

Here, \( u = x^2 - 2x \), so:

\[
\frac{du}{dx} = 2x - 2.
\]

Thus, applying the chain rule:

\[
\frac{d}{dx} \left[ f(x^2 - 2x) \right] = f'(x^2 - 2x) \cdot (2x - 2).
\]

### Step 2: Set up the given equation

From the problem, we are told that:

\[
\frac{d}{dx} \left[ f(x^2 - 2x) \right] = x^2 - 3x + 2.
\]

Equating the two expressions:

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

### Step 3: Solve for \( f'(g(x)) \)

Since \( g(x) = x^2 - 2x \), we are tasked with finding \( f'(g(x)) \), which is the same as \( f'(x^2 - 2x) \). To isolate \( f'(x^2 - 2x) \), divide both sides of the equation by \( (2x - 2) \):

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

### Step 4: Simplify the right-hand side

The expression \( \frac{x^2 - 3x + 2}{2x - 2} \) can be simplified by factoring both the numerator and denominator:

- The numerator \( x^2 - 3x + 2 \) factors as \( (x - 1)(x - 2) \).
- The denominator \( 2x - 2 \) factors as \( 2(x - 1) \).

So, the expression becomes:

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

Cancel the \( (x - 1) \) terms:

\[
f'(x^2 - 2x) = \frac{x - 2}{2}.
\]

Thus, \( f'(g(x)) = \frac{x - 2}{2} \).

### Final Answer:

\[
f'(g(x)) = \frac{x - 2}{2}.
\]

Let me know if you'd like more details or explanations on any step! 

Here are some related questions for practice:

1. What is the chain rule, and how is it applied in this problem?
2. Can you simplify the expression \( \frac{x^2 - 3x + 2}{2x - 2} \) in a different way?
3. How would you differentiate a composite function like \( f(g(x)) \) if both functions were more complicated?
4. What happens if you substitute specific values of \( x \) into \( f'(g(x)) \)?
5. How does factoring help in simplifying the given expression for \( f'(g(x)) \)?

**Tip:** Always try to simplify algebraic expressions, especially in problems involving derivatives, to make the calculation easier.

Find f'(g(x)) if d/dx [f(x^2 - 2x)] = x^2 - 3x + 2 and g(x) = x^2 - 2x.

Learn how to find f'(g(x)) using the chain rule in differentiation. This problem involves algebraic simplifications and factoring. Suitable for advanced high school or introductory calculus students.

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