Chain Rule Application Sequence

To differentiate the function $f(x)=\cos(\ln(x^2+1))$ using the chain rule, the following steps are performed. What is the correct logical sequence for these steps?

A: Differentiate the outer function, noting that $\frac{d}{du}[\cos(u)] = -\sin(u)$.
B: Identify the inner function as $u = \ln(x^2+1)$.
C: Differentiate the inner function to find $\frac{du}{dx} = \frac{2x}{x^2+1}$.
D: Multiply the results to obtain the final derivative, $-\frac{2x\sin(\ln(x^2+1))}{x^2+1}$