Differentiate $$f(x)=\cos(\ln(x^2+1))$$ using the chain rule. Rank the following steps from outermost to innermost:
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 obtain $$\frac{2*x}{x^2+1}$$.
D: Multiply the results to obtain the final derivative $$-\frac{2*x*sin(\ln(x^2+1))}{x^2+1}$$.