How do I differentiate sin(x²) using the Chain Rule? Step-by-step help needed

Hi everyone! 👋

I am a second-year calculus student and I am struggling to understand how to apply the Chain Rule correctly. My professor gave us this problem and I cannot figure out the steps.

The function is:

$$f(x) = \sin(x^2)$$

I know the Chain Rule formula is:

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

But I am confused about how to identify the outer function and the inner function here.

Here is what I tried so far:

I said the outer function is $f(u) = \sin(u)$ and the inner function is $g(x) = x^2$

Then I got:

$$f'(x) = \cos(x^2) \cdot 2x$$

Is this correct? And can someone explain WHY we multiply by $2x$ at the end? I understand the formula but not the intuition behind it.

Also, how would this change if the function was $\sin(x^3)$ or $\sin(2x+1)$?

Any help would be really appreciated. Thank you! 🙏