find the derivative of y with respect to x.\ny = ln(16x) + 4x\n\\frac{dy}{dx} = \\square

Explanation:

Step1: Differentiate $\ln(16x)$

Use the chain - rule. If (y = \ln(u)) and (u = 16x), then (\frac{dy}{du}=\frac{1}{u}) and (\frac{du}{dx}=16). By the chain - rule (\frac{d}{dx}(\ln(16x))=\frac{1}{16x}\times16=\frac{1}{x}).

Step2: Differentiate (4x)

Use the power - rule (\frac{d}{dx}(ax^{n})=anx^{n - 1}). For (y = 4x) (where (a = 4) and (n = 1)), (\frac{d}{dx}(4x)=4\times1\times x^{1-1}=4).

Step3: Sum the derivatives

By the sum - rule of differentiation (\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}(f(x))+\frac{d}{dx}(g(x))). Here (f(x)=\ln(16x)) and (g(x)=4x), so (\frac{dy}{dx}=\frac{1}{x}+4).

Answer:

(\frac{1}{x}+4)