determine the derivative of the function (g(x)=(ln x)^{24})

Explanation:

Step1: Use the chain - rule.

Let $u = \ln x$, then $g(x)=u^{24}$. The chain - rule states that if $y = f(u)$ and $u = h(x)$, then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{d}{du}(u^{24})$. $\frac{d}{du}(u^{24}) = 24u^{23}$

Step2: Find $\frac{du}{dx}$.

Since $u=\ln x$, $\frac{du}{dx}=\frac{1}{x}$

Step3: Apply the chain - rule.

$\frac{dg}{dx}=\frac{d}{du}(u^{24})\cdot\frac{du}{dx}=24u^{23}\cdot\frac{1}{x}$ Substitute $u = \ln x$ back in. $\frac{dg}{dx}=\frac{24(\ln x)^{23}}{x}$

Answer:

$\frac{24(\ln x)^{23}}{x}$