To find the derivative of $\cos(x^2)$ with respect to $x$, we use the chain rule.

Given: $f(x) = \cos(x^2)$

1. Let $u = x^2$. Then $f(x) = \cos(u)$.
2. Find $\frac{du}{dx}$: $\frac{du}{dx} = 2x$
3. Now, differentiate $f(u) = \cos(u)$ with respect to $u$: $\frac{d}{du} \cos(u) = -\sin(u)$
4. Using the chain rule: $\frac{d}{dx} \cos(x^2) = \frac{d}{du} \cos(u) \cdot \frac{du}{dx}$ Substituting, we get: $\frac{d}{dx} \cos(x^2) = -\sin(x^2) \cdot 2x$
5. Simplify: $\frac{d}{dx} \cos(x^2) = -2x \sin(x^2)$

So, the derivative of $\cos(x^2)$ with respect to $x$ is: $\frac{d}{dx} \cos(x^2) = -2x \sin(x^2)$

Would you like further details or have any questions?

Here are some related questions you might consider:

1. How do we apply the chain rule in more complex expressions?
2. What would be the derivative of $\sin(x^2)$?
3. How does this derivative change if we consider $\cos(kx^2)$ for a constant $k$?
4. What if we need to differentiate $\cos(x^3)$?
5. How would we find the second derivative of $\cos(x^2)$?

Tip: Always identify the "inner" and "outer" functions separately when using the chain rule.