Question

the derivative of ( f(x)=12 ln sqrt3{1-2 x} ) is ( f^{prime}(x)=\frac{c}{2 x-1} ). find the value of ( c ). give an exact answer as an integer.

Explanation:

Step1: Simplify the function

First, simplify \(f(x)=12\ln\sqrt[3]{1 - 2x}\). Using the property \(\ln a^b=b\ln a\), we have \(f(x)=12\times\frac{1}{3}\ln(1 - 2x)=4\ln(1 - 2x)\).

Step2: Differentiate using the chain rule

The derivative of \(\ln u\) with respect to \(x\) is \(\frac{u'}{u}\) (chain rule, where \(u = 1-2x\) and \(u'=- 2\)).
So \(f'(x)=\frac{4\times(-2)}{1 - 2x}=\frac{-8}{1 - 2x}=\frac{8}{2x - 1}\).

Answer:

\(c = 8\)