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.

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.