find the derivative of y with respect to x.\ny = log 3x + log 3x³\ndy/dx = □

Explanation:

Step1: Simplify the function

Use the logarithmic property (\log_a b^n=n\log_a b). For (y = \log_3x+\log_3x^3), since (\log_3x^3 = 3\log_3x), then (y=\log_3x + 3\log_3x=4\log_3x). Also, use the change - of - base formula (\log_a x=\frac{\ln x}{\ln a}). So (y = 4\times\frac{\ln x}{\ln 3}=\frac{4}{\ln 3}\ln x).

Step2: Differentiate the function

Use the formula (\frac{d}{dx}(\ln x)=\frac{1}{x}). Since (y=\frac{4}{\ln 3}\ln x), and (\frac{4}{\ln 3}) is a constant. By the constant - multiple rule ((cf(x))^\prime = cf^\prime(x)) where (c = \frac{4}{\ln 3}) and (f(x)=\ln x). Then (\frac{dy}{dx}=\frac{4}{\ln 3}\times\frac{1}{x}).

Answer:

(\frac{4}{x\ln 3})