Question

find f(x). f(x)=(9 + ln x)^3 f(x)=□

Explanation:

Step1: Identify the outer - inner functions

Let $u = 9+\ln x$, so $y = u^{3}$.

Step2: Differentiate the outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=3u^{2}$. Substituting $u = 9+\ln x$ back in, we get $\frac{dy}{du}=3(9 + \ln x)^{2}$.

Step3: Differentiate the inner function

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

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. So $f^{\prime}(x)=3(9 + \ln x)^{2}\cdot\frac{1}{x}=\frac{3(9+\ln x)^{2}}{x}$.

Answer:

$\frac{3(9+\ln x)^{2}}{x}$