To find the derivative of the function \( f(x) = \cos(x) \sqrt{1 + x^2} \), we will use the product rule, since the function is a product of two functions:

\[
f(x) = u(x) \cdot v(x), \quad \text{where} \quad u(x) = \cos(x), \, v(x) = \sqrt{1 + x^2}.
\]

### Step-by-Step Solution

#### 1. Apply the Product Rule

The product rule states that:

\[
(f \cdot g)' = f' \cdot g + f \cdot g'.
\]

So, the derivative of \( f(x) = u(x) \cdot v(x) \) is:

\[
f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x).
\]

Now, we need to find \( u'(x) \) and \( v'(x) \).

#### 2. Find the Derivatives of \( u(x) \) and \( v(x) \)

- \( u(x) = \cos(x) \), so:

\[
u'(x) = -\sin(x).
\]

- \( v(x) = \sqrt{1 + x^2} = (1 + x^2)^{1/2} \), so:

\[
v'(x) = \frac{1}{2} (1 + x^2)^{-1/2} \cdot (2x) = \frac{x}{\sqrt{1 + x^2}}.
\]

#### 3. Substitute and Simplify

Substitute the derivatives into the product rule formula:

\[
f'(x) = (-\sin(x)) \cdot \sqrt{1 + x^2} + \cos(x) \cdot \frac{x}{\sqrt{1 + x^2}}.
\]

Simplify the expression:

\[
f'(x) = -\sin(x) \sqrt{1 + x^2} + \frac{x \cos(x)}{\sqrt{1 + x^2}}.
\]

#### 4. Combine into a Single Fraction

To combine these terms into a single fraction, find a common denominator:

\[
f'(x) = \frac{-\sin(x) (1 + x^2) + x \cos(x)}{\sqrt{1 + x^2}}.
\]

#### 5. Final Answer

So, the derivative of \( f(x) = \cos(x) \sqrt{1 + x^2} \) is:

\[
\boxed{f'(x) = \frac{-\sin(x) (1 + x^2) + x \cos(x)}{\sqrt{1 + x^2}}.}
\]

### Questions for Further Exploration:

1. What would the derivative be if the function were \( \sin(x) \sqrt{1 + x^2} \)?
2. How does the product rule apply to functions that are multiplied by a constant?
3. Can you find the second derivative of the function \( f(x) = \cos(x) \sqrt{1 + x^2} \)?
4. How would you apply the chain rule to a more complex function involving trigonometric and square root terms?
5. What is the geometric interpretation of the derivative of this function?

### Tip:

When dealing with the product of functions, always remember to apply the product rule, and simplify the resulting expression carefully to avoid common mistakes.

Explore the step-by-step solution to differentiate cos(x)√(1 + x^2) using the product and chain rules. This calculus problem covers essential trigonometric and differentiation concepts. Suitable for advanced high school students or undergraduates.

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