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We are given the equation:

\[
f(x) + x^2 [f(x)]^3 = 10
\]
with the condition that \( f(1) = 2 \), and we need to find \( f'(1) \).

### Step-by-step solution:

1. **Differentiate both sides of the equation** with respect to \(x\), using implicit differentiation. Recall that \(f(x)\) is a function of \(x\), so we will need to apply the chain rule.

   The left-hand side is:
   \[
   \frac{d}{dx} \left( f(x) + x^2 [f(x)]^3 \right)
   \]
   Differentiate term by term:

   - For \(f(x)\), the derivative is simply \(f'(x)\).
   
   - For the second term, \(x^2 [f(x)]^3\), apply the product rule:
     \[
     \frac{d}{dx} \left( x^2 [f(x)]^3 \right) = \frac{d}{dx} \left( x^2 \right) [f(x)]^3 + x^2 \frac{d}{dx} \left( [f(x)]^3 \right)
     \]
     The first term becomes:
     \[
     2x [f(x)]^3
     \]
     For the second part, apply the chain rule to \( [f(x)]^3 \):
     \[
     x^2 \cdot 3[f(x)]^2 f'(x)
     \]
   
   So, the total derivative of the left-hand side is:
   \[
   f'(x) + 2x [f(x)]^3 + 3x^2 [f(x)]^2 f'(x)
   \]

2. **Differentiate the right-hand side**, which is a constant (10). Its derivative is zero.

   Therefore, the differentiated equation is:
   \[
   f'(x) + 2x [f(x)]^3 + 3x^2 [f(x)]^2 f'(x) = 0
   \]

3. **Substitute \(x = 1\) and \(f(1) = 2\)** into the differentiated equation:
   \[
   f'(1) + 2(1) [2]^3 + 3(1)^2 [2]^2 f'(1) = 0
   \]
   Simplifying the terms:
   \[
   f'(1) + 2(8) + 3(4) f'(1) = 0
   \]
   This becomes:
   \[
   f'(1) + 16 + 12 f'(1) = 0
   \]
   Combine like terms:
   \[
   13 f'(1) + 16 = 0
   \]

4. **Solve for \(f'(1)\):**
   \[
   13 f'(1) = -16
   \]
   \[
   f'(1) = -\frac{16}{13}
   \]

Thus, the derivative \(f'(1)\) is:
\[
f'(1) = -\frac{16}{13}
\]

---

Would you like any more details or clarifications? 

Here are some related questions you might find helpful:

1. How does implicit differentiation work for more complex functions?
2. Can you explain the product rule in detail?
3. How do we use the chain rule in composite functions like \([f(x)]^3\)?
4. What are common mistakes to avoid when differentiating implicitly?
5. How can we verify solutions for implicitly defined functions?

**Tip:** Always double-check chain rule applications, especially when differentiating powers of functions!

If f(x) + x^2 [f(x)]^3 = 10 and f(1) = 2, find f'(1).

Learn how to find f'(1) from the equation f(x) + x^2 [f(x)]^3 = 10 with the condition f(1) = 2. This calculus problem involves implicit differentiation, the product rule, and the chain rule. Step-by-step solution provided, ideal for students in Grades 11-12.

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