find the derivative of the following function.\n$y = \\ln(3x^{5})$\nanswer\n$y=$

Explanation:

Step1: Use the logarithm property $\ln(ab)=\ln a+\ln b$

$$y = \ln(3x^{5})=\ln3+\ln(x^{5})$$ Since $\ln3$ is a constant, its derivative is $0$. And for $\ln(x^{5})$, use the property $\ln(a^{b})=b\ln a$, so $\ln(x^{5}) = 5\ln x$.

Step2: Differentiate $y = 5\ln x$

The derivative of $\ln x$ is $\frac{1}{x}$. Using the constant - multiple rule $(cf(x))^\prime = cf^\prime(x)$ (where $c = 5$ and $f(x)=\ln x$), we have $y^\prime=5\times\frac{1}{x}$.

Answer:

$\frac{5}{x}$