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The question in the image involves calculating the first derivative of the function \( f(x) = \cos(x^2) + e^x + 3 \), and selecting the correct derivative from a set of multiple-choice alternatives.

### Given function:
\[ f(x) = \cos(x^2) + e^x + 3 \]

### Applying derivative rules:
1. **Derivative of \( \cos(x^2) \)**:
   - Chain rule: \( \frac{d}{dx}[\cos(g(x))] = -\sin(g(x)) \cdot g'(x) \), where \( g(x) = x^2 \).
   - So, \( \frac{d}{dx}[\cos(x^2)] = -\sin(x^2) \cdot 2x \).

2. **Derivative of \( e^x \)**:
   - The derivative of \( e^x \) is simply \( e^x \).

3. **Derivative of 3**:
   - The derivative of a constant is 0.

### Therefore, the derivative of \( f(x) \) is:
\[ f'(x) = -2x \sin(x^2) + e^x \]

Now, let's check the alternatives from the image to find the matching one:

- **Option A**: \( f'(x) = \cos(x^2) + e^x \) — Incorrect (cosine term should change).
- **Option B**: \( f'(x) = -\sin(x) + e^x \) — Incorrect (the argument of sine is incorrect).
- **Option C**: \( f'(x) = \cos(x) + e^x \) — Incorrect.
- **Option D**: \( f'(x) = -2x \sin(x^2) + e^x \) — Correct.
- **Option E**: \( f'(x) = -\sin(x^2) + e^x \) — Incorrect (missing the \( 2x \) factor).

### Correct answer: **Option D**.

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Do you need more details on the steps? 

Here are some related questions to deepen your understanding:
1. What is the chain rule, and how is it applied to composite functions like \( \cos(x^2) \)?
2. How do you handle the derivative of an exponential function such as \( e^x \)?
3. Can you explain why the derivative of a constant is zero?
4. What would be the second derivative of the function \( f(x) = \cos(x^2) + e^x + 3 \)?
5. How do you differentiate trigonometric functions involving composite arguments?

**Tip**: Always check if the function involves composite expressions where the chain rule might apply.

Uma das maneiras de obter a derivada de uma função é pela definição. Porém, dependendo da função, esse processo pode ser bastante longo e complexo. Em contrapartida, existem regras de derivação que facilitam todo esse processo. Com base nas regras de derivação, analise a função a seguir: f(x) = cos(x^2) + e^x + 3. Assinale a alternativa que indica a derivada de 1ª ordem da função f: a) f'(x) = cos(x^2) + e^x; b) f'(x) = -sin(x) + e^x; c) f'(x) = cos(x) + e^x; d) f'(x) = -2x sin(x^2) + e^x; e) f'(x) = -sin(x^2) + e^x

Learn how to find the first derivative of the function f(x) = cos(x^2) + e^x + 3. This problem covers concepts of trigonometric and exponential derivatives, with a focus on applying the chain rule and basic differentiation. Suitable for undergraduate calculus students.

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