Question

find the derivative of y with respect to x.
y = log 3x + log 3x³
dy/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:

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