Question

find the derivative of y with respect to x.

$y = 2 ln left( \frac { 8 } { x }
ight)$

$\frac { d y } { d x } = square$

Explanation:

Step1: Simplify the function

Use the logarithm property \(\ln\frac{a}{b}=\ln a-\ln b\). So \(y = 2\ln8-2\ln x\). Since \(\ln8\) is a constant, the derivative of \(2\ln8\) with respect to \(x\) is \(0\). Now we only need to find the derivative of \(- 2\ln x\).

Step2: Apply the derivative formula

The derivative of \(\ln x\) with respect to \(x\) is \(\frac{1}{x}\). Using the constant - multiple rule \((cf(x))^\prime = cf^\prime(x)\) (where \(c=-2\) and \(f(x)=\ln x\)), we have \((-2\ln x)^\prime=-2\times\frac{1}{x}\).

Answer:

\(-\frac{2}{x}\)