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, $$-\frac{2x\sin(\ln(x^2+1))}{x^2+1}$$