1. **State the problem:** Find the derivative $\frac{dy}{dx}$ if $y = (\cos x)^{\sin x}$ using logarithmic differentiation. 2. **Take the natural logarithm of both sides:** $$\ln y = \ln \left((\cos x)^{\sin x}\right)$$ Using the logarithm power rule: $$\ln y = \sin x \cdot \ln (\cos x)$$ 3. **Differentiate both sides with respect to $x$:** Using implicit differentiation on the left and product rule on the right: $$\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} (\sin x \cdot \ln (\cos x))$$ 4. **Apply the product rule:** $$\frac{d}{dx} (\sin x \cdot \ln (\cos x)) = \cos x \cdot \ln (\cos x) + \sin x \cdot \frac{d}{dx} (\ln (\cos x))$$ 5. **Differentiate $\ln (\cos x)$:** $$\frac{d}{dx} (\ln (\cos x)) = \frac{1}{\cos x} \cdot (-\sin x) = -\tan x$$ 6. **Substitute back:** $$\frac{1}{y} \frac{dy}{dx} = \cos x \ln (\cos x) - \sin x \tan x$$ 7. **Multiply both sides by $y$ to solve for $\frac{dy}{dx}$:** $$\frac{dy}{dx} = y \left(\cos x \ln (\cos x) - \sin x \tan x\right)$$ 8. **Recall that $y = (\cos x)^{\sin x}$:** $$\frac{dy}{dx} = (\cos x)^{\sin x} \left(\cos x \ln (\cos x) - \sin x \tan x\right)$$ **Final answer:** $$\boxed{\frac{dy}{dx} = (\cos x)^{\sin x} \left(\cos x \ln (\cos x) - \sin x \tan x\right)}$$