1. **State the problem:** We need to find the derivative of the function $f(x) = 2x(1 - \ln x)$. 2. **Recall the formula:** To differentiate a product of two functions, use the product rule: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ where $u(x) = 2x$ and $v(x) = 1 - \ln x$. 3. **Find derivatives of each part:** - $u'(x) = \frac{d}{dx}[2x] = 2$ - $v'(x) = \frac{d}{dx}[1 - \ln x] = 0 - \frac{1}{x} = -\frac{1}{x}$ 4. **Apply the product rule:** $$f'(x) = u'(x)v(x) + u(x)v'(x) = 2(1 - \ln x) + 2x\left(-\frac{1}{x}\right)$$ 5. **Simplify:** $$f'(x) = 2 - 2\ln x - 2 = -2\ln x$$ 6. **Final answer:** $$\boxed{f'(x) = -2\ln x}$$