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

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

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