Question

find the derivative of y with respect to x.
y = log₆x + log₆x³
dy/dx = □

Explanation:

Step1: Simplify the function

Use the logarithm property $\log_a M^n=n\log_a M$. So, $y = \log_6x+\log_6x^3=\log_6x + 3\log_6x=4\log_6x$.

Step2: Recall the derivative formula

The derivative of $\log_a x$ is $\frac{1}{x\ln a}$. For $y = 4\log_6x$, by the constant - multiple rule of differentiation $(cf(x))'=cf'(x)$ where $c = 4$ and $f(x)=\log_6x$.
The derivative of $\log_6x$ is $\frac{1}{x\ln 6}$, so $y'=\frac{4}{x\ln 6}$.

Answer:

$\frac{4}{x\ln 6}$