find f(x). f(x)=(2 + ln x)^8 f(x)=

Explanation:

Step1: Identify the outer - inner functions

Let $u = 2+\ln x$, then $y = u^{8}$.

Step2: Differentiate the outer function

Using the power rule $\frac{d}{du}(u^{n})=nu^{n - 1}$, for $y = u^{8}$, we have $\frac{dy}{du}=8u^{7}$.

Step3: Differentiate the inner function

Since $u = 2+\ln x$, then $\frac{du}{dx}=\frac{1}{x}$.

Step4: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $\frac{dy}{du}=8u^{7}$ and $\frac{du}{dx}=\frac{1}{x}$ and $u = 2+\ln x$ back in, we get $\frac{dy}{dx}=8(2 + \ln x)^{7}\cdot\frac{1}{x}$.

Answer:

$\frac{8(2+\ln x)^{7}}{x}$