Chain Rule For A Composite Function

Identify the outer function $\cos(u)$ and inner function $\ln(x)$, differentiate the outer function to get -sin(ln(x)), differentiate the inner function to get 1/x, then multiply these derivatives

Differentiate $\cos(\ln(x))$ by applying the product rule between $\cos$ and $\ln(x)$, then multiply by 1

Identify the outer function as $\ln(x)$ and the inner function as $\cos(u)$, then differentiate in that order and multiply

Differentiate $\ln(x)$ first treating it as a constant, then differentiate $\cos(u)$ without applying the chain rule properly, and finally multiply