differentiate. f(x) = ln(6x) f(x) =

Explanation:

Step1: Recall chain - rule

The chain - rule states that if $y = f(g(x))$, then $y^\prime=f^\prime(g(x))\cdot g^\prime(x)$. Let $u = 6x$, so $y=\ln(u)$.

Step2: Differentiate outer function

The derivative of $y = \ln(u)$ with respect to $u$ is $\frac{dy}{du}=\frac{1}{u}$.

Step3: Differentiate inner function

The derivative of $u = 6x$ with respect to $x$ is $\frac{du}{dx}=6$.

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=\frac{1}{u}$ and $\frac{du}{dx}=6$ into the chain - rule formula. Since $u = 6x$, we have $\frac{dy}{dx}=\frac{1}{6x}\cdot6$.

Step5: Simplify the result

$\frac{1}{6x}\cdot6=\frac{1}{x}$.

Answer:

$\frac{1}{x}$