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Let's tackle the given problem step by step.

### A. Compute the derivative of \( f(x) = \sin(x^2) \).

We will apply the **chain rule** to compute the derivative of \( f(x) = \sin(x^2) \).

1. The outer function is \( \sin(u) \), and its derivative is \( \cos(u) \), where \( u = x^2 \).
2. The inner function is \( u = x^2 \), and its derivative is \( 2x \).

By the chain rule, the derivative of \( f(x) \) is:
\[
f'(x) = \frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot \frac{d}{dx}(x^2) = \cos(x^2) \cdot 2x
\]
Thus, the derivative of \( f(x) \) is:
\[
f'(x) = 2x \cos(x^2)
\]

---

### B. Let \( F(x) \) be a function such that \( F(0) = 0 \) and \( F'(x) = f(x) = \sin(x^2) \). Sketch the graph of \( F(x) \).

In this part, we are asked to sketch the graph of \( F(x) \), where \( F'(x) = \sin(x^2) \) and \( F(0) = 0 \).

To understand \( F(x) \), note that:
- \( F(x) \) is the **antiderivative** of \( \sin(x^2) \), meaning it is an integral of the form:
  \[
  F(x) = \int_0^x \sin(t^2) \, dt
  \]
  This is related to **Fresnel integrals**, which are known to be non-elementary (they cannot be expressed in terms of elementary functions).

- Although the integral does not have a simple closed form, we can still describe general properties of the graph:
  - At \( x = 0 \), \( F(0) = 0 \), which is a given condition.
  - Since \( \sin(x^2) \) oscillates between -1 and 1, the function \( F(x) \) will oscillate but grow slowly over time as the oscillations become smaller for large values of \( x \).

To sketch the graph:
- The curve will start at \( (0, 0) \).
- As \( x \) increases, the function \( F(x) \) oscillates as the sine term oscillates but with decreasing amplitude as \( x^2 \) grows. The curve will look like an oscillating function that gets flatter over time, similar to the behavior of Fresnel integrals.

Would you like to dive deeper into these topics or explore further with a more detailed explanation of the Fresnel integrals?

### Related Questions:
1. What is the chain rule in calculus, and how does it apply to this problem?
2. Can you explain why the function \( \sin(x^2) \) does not have an elementary antiderivative?
3. What are Fresnel integrals, and how are they related to this problem?
4. What general properties can we infer about the behavior of \( F(x) \) for large \( x \)?
5. How does the oscillatory nature of sine affect the integral and graph of \( F(x) \)?

### Tip:
When dealing with functions that involve complex integrals, visualizing the graph and understanding the behavior through properties (such as oscillation or limits) can be helpful even when a closed-form solution is not possible.

Let f(x) = sin(x^2). A. Compute the derivative of f. B. Let F(x) be a function such that F(0) = 0 and F'(x) = f(x). Sketch the graph of F(x).

Learn how to compute the derivative of f(x) = sin(x^2) using the chain rule. Additionally, explore the behavior of the antiderivative F(x) and how it relates to Fresnel integrals. This problem involves complex integrals and is suitable for undergraduate students.

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