Question

but be aware: this assignment is set up to be auto - graded so you have immediate feedback.
be exactly the same. on in - class assessments, you will be asked to write sentences and/or sl
these types of problems.
1 multiple choice 1 point
what is the derivative of $\frac{6}{x^{3}}$?
$6\ln(x^{3})$
$-18x^{-4}$
$\frac{1}{3x^{2}}$
$6x^{-3}$
$18x^{2}$
$-18x^{-2}$
clear my selection

Explanation:

Step1: Rewrite the function

Rewrite \(\frac{6}{x^{3}}\) as \(6x^{- 3}\) using the rule \(\frac{a}{x^{n}}=ax^{-n}\).

Step2: Apply the power rule

The power rule for differentiation is \(\frac{d}{dx}(x^{n})=nx^{n - 1}\). For \(y = 6x^{-3}\), using the constant - multiple rule \(\frac{d}{dx}(cf(x))=c\frac{d}{dx}(f(x))\) (where \(c = 6\) and \(f(x)=x^{-3}\)), we have \(\frac{d}{dx}(6x^{-3})=6\frac{d}{dx}(x^{-3})\).
Then, by the power rule, \(6\times(-3)x^{-3 - 1}\).

Step3: Simplify the expression

\(6\times(-3)x^{-3 - 1}=-18x^{-4}\)

Answer:

\(-18x^{-4}\) (the second option)