Let's solve the problem $\frac{dy}{dx}$ for $y = \sqrt{1 + \cos(x^2)}$.

Solution:

The function given is: $y = \sqrt{1 + \cos(x^2)}$

Step 1: Rewrite the square root

Rewrite $y$ as: $y = \left(1 + \cos(x^2)\right)^{1/2}$

Step 2: Apply the chain rule

Differentiate both sides with respect to $x$: $\frac{dy}{dx} = \frac{1}{2} \left(1 + \cos(x^2)\right)^{-1/2} \cdot \frac{d}{dx}(1 + \cos(x^2))$

Step 3: Differentiate $1 + \cos(x^2)$

The derivative of $1$ is $0$, and the derivative of $\cos(x^2)$ is: $\frac{d}{dx} \cos(x^2) = -\sin(x^2) \cdot \frac{d}{dx}(x^2) = -\sin(x^2) \cdot 2x$

So: $\frac{d}{dx}(1 + \cos(x^2)) = -2x \sin(x^2)$

Step 4: Substitute back

Substituting this into the expression for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{1}{2} \left(1 + \cos(x^2)\right)^{-1/2} \cdot (-2x \sin(x^2))$

Simplify: $\frac{dy}{dx} = \frac{-x \sin(x^2)}{\sqrt{1 + \cos(x^2)}}$

Final Answer:

$\boxed{\frac{dy}{dx} = \frac{-x \sin(x^2)}{\sqrt{1 + \cos(x^2)}}}$

Let me know if you'd like a more detailed explanation or have questions!

Here are 5 related questions to explore further:

1. How do you apply the chain rule in similar problems with nested functions?
2. What is the derivative of $y = \sqrt{1 - \cos(x^2)}$?
3. How would the derivative change if $y = \sqrt{1 + \sin(x^2)}$?
4. Can you explain why $\frac{d}{dx} \cos(x^2)$ includes $\frac{d}{dx}(x^2)$?
5. What are some common applications of derivatives involving square roots in physics or engineering?

Tip: When working with square root functions, rewriting them as powers (e.g., $\sqrt{u}$ as $u^{1/2}$) makes differentiation easier!