Question

the derivative of ( f(x)=\frac{ln (6 x+5)}{3} ) is ( f^{prime}(x)=\frac{c}{6 x+5} ).
find the value of ( c ).
give an exact answer as an integer.

Explanation:

Step1: Differentiate \(f(x)=\frac{\ln(6x + 5)}{3}\)

Use the chain - rule \((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}\) (here \(v = 3\), \(u=\ln(6x + 5)\)). The derivative of \(\ln(u)\) with respect to \(x\) is \(\frac{u^\prime}{u}\), where \(u = 6x+5\) and \(u^\prime=6\). So \(f^\prime(x)=\frac{1}{3}\times\frac{6}{6x + 5}\).

Step2: Compare with \(f^\prime(x)=\frac{c}{6x + 5}\)

Since \(f^\prime(x)=\frac{1}{3}\times\frac{6}{6x + 5}=\frac{2}{6x + 5}\), and \(f^\prime(x)=\frac{c}{6x + 5}\).

Answer:

\(2\)