what is the derivative of 4 cos(x) - ln(x)?\n4 sin(x) - x ln(x)\nsin(4x) - \\frac{1}{x}\n- sin(4x)…

Explanation:

Step1: Differentiate (4\cos(x))

Using the rule (\frac{d}{dx}(a\cos(x))=-a\sin(x)), for (a = 4), we have (\frac{d}{dx}(4\cos(x))=-4\sin(x))

Step2: Differentiate (-\ln(x))

Using the rule (\frac{d}{dx}(\ln(x))=\frac{1}{x}), so (\frac{d}{dx}(-\ln(x))=-\frac{1}{x})

Step3: Combine the derivatives

By the sum - rule of differentiation (\frac{d}{dx}(u + v)=\frac{du}{dx}+\frac{dv}{dx}), if (u = 4\cos(x)) and (v=-\ln(x)), then (\frac{d}{dx}(4\cos(x)-\ln(x))=-4\sin(x)-\frac{1}{x})

Answer:

(-4\sin(x)-\frac{1}{x}) (the fourth option)