determine the derivative of the function g(x) = (lnx)^14.

Explanation:

Step1: Identify the outer - inner functions

Let $u = \ln x$, so $g(x)=u^{14}$.

Step2: Differentiate the outer function

The derivative of $y = u^{14}$ with respect to $u$ is $\frac{dy}{du}=14u^{13}$.

Step3: Differentiate the inner function

The derivative of $u=\ln x$ with respect to $x$ is $\frac{du}{dx}=\frac{1}{x}$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dg}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = \ln x$, $\frac{dy}{du}=14u^{13}$ and $\frac{du}{dx}=\frac{1}{x}$ into the chain - rule formula. We get $\frac{dg}{dx}=14(\ln x)^{13}\cdot\frac{1}{x}$.

Answer:

$\frac{14(\ln x)^{13}}{x}$