Question

find the derivative of y with respect to x.
y = 3 ln (\frac{9}{x})
\frac{dy}{dx}=\square

Explanation:

Step1: Use log - property

We know that $\ln(\frac{a}{b})=\ln(a)-\ln(b)$. So, $y = 3\ln(\frac{9}{x})=3(\ln(9)-\ln(x))=3\ln(9)-3\ln(x)$.

Step2: Differentiate term - by - term

The derivative of a constant $C$ with respect to $x$ is $0$. Since $3\ln(9)$ is a constant, its derivative is $0$. The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$. Using the constant - multiple rule $\frac{d}{dx}(kf(x))=k\frac{d}{dx}(f(x))$, where $k = - 3$ and $f(x)=\ln(x)$. The derivative of $-3\ln(x)$ is $-3\times\frac{1}{x}=-\frac{3}{x}$.
So, $\frac{dy}{dx}=0-\frac{3}{x}=-\frac{3}{x}$.

Answer:

$-\frac{3}{x}$