If a(x) = 2x^3+1, e(x) = x^2 then d`(e(3)) = zero or 9 or 486 or 2916

To solve the problem, we need to find \( d'(e(3)) \). Let's break down the steps:

1. **Identify \( e(3) \):**
   - The function \( e(x) = x^2 \).
   - Therefore, \( e(3) = 3^2 = 9 \).

2. **Differentiate \( a(x) \) to find \( a'(x) \):**
   - The function \( a(x) = 2x^3 + 1 \).
   - The derivative \( a'(x) = \frac{d}{dx}(2x^3 + 1) \).
   - Differentiating, we get:
     \[
     a'(x) = 6x^2
     \]

3. **Evaluate \( a'(x) \) at \( x = e(3) \):**
   - \( a'(e(3)) = a'(9) \).
   - Substitute \( x = 9 \) into \( a'(x) = 6x^2 \):
     \[
     a'(9) = 6(9)^2 = 6 \times 81 = 486
     \]

So, \( d'(e(3)) = a'(9) = 486 \).

### Answer: 
The correct answer is **486**.

Would you like more details or have any other questions?

### Follow-up Questions:
1. How would the result change if \( e(x) = 3x^2 \) instead of \( x^2 \)?
2. What is the physical interpretation of a derivative in this context?
3. Can we generalize \( d'(e(x)) \) for any \( x \) given the functions \( a(x) \) and \( e(x) \)?
4. What would \( d'(e(3)) \) be if \( a(x) = x^3 + x \)?
5. How does the chain rule apply to this problem?

### Tip:
When differentiating composite functions, always apply the chain rule, which states \( d(f(g(x)))/dx = f'(g(x)) \cdot g'(x) \).

Learn how to solve the problem of finding d'(e(3)) for the given functions a(x) = 2x^3 + 1 and e(x) = x^2 using differentiation. This problem covers key calculus concepts, including the chain rule and composite functions. Suitable for advanced high school or college calculus students.

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