https://file.aiquickdraw.com/mathbot/file/17265107204125d51rj5s.jpg

Let's analyze and solve the problem from the image step by step. The problem states:

Let \( f \) be a function defined on \( \mathbb{R} \) by:
\[
f(x) = x \sqrt{x^2 + 4}
\]
We are asked to demonstrate that for any real \( x \), the derivative of \( f(x) \), denoted \( f'(x) \), is:
\[
f'(x) = \frac{2(x^2 + 2)}{\sqrt{x^2 + 4}}
\]

### Step 1: Apply the product and chain rule

We need to differentiate \( f(x) = x \sqrt{x^2 + 4} \). This is a product of two functions:
- \( u(x) = x \)
- \( v(x)

Soit f la fonction définie sur ℝ par f(x) = x√(x² + 4). Démontrer que pour tout réel x, f'(x) = 2(x² + 2)/√(x² + 4).

Explore a detailed step-by-step solution to differentiate f(x) = x√(x² + 4). This problem applies the product rule and chain rule for differentiation and provides a complete analysis of the derivative for any real number x. Ideal for undergraduate calculus students.

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