Question

find f(x).
f(x)=(5 + ln x)^6
f(x)=□

Explanation:

Step1: Identify the outer - inner functions

Let $u = 5+\ln x$, then $y = u^{6}$.

Step2: Differentiate the outer function

The derivative of $y = u^{6}$ with respect to $u$ is $\frac{dy}{du}=6u^{5}$.

Step3: Differentiate the inner function

The derivative of $u = 5+\ln x$ with respect to $x$ is $\frac{du}{dx}=\frac{1}{x}$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 5+\ln x$, $\frac{dy}{du}=6u^{5}$ and $\frac{du}{dx}=\frac{1}{x}$ into the chain - rule formula. We get $\frac{dy}{dx}=6(5 + \ln x)^{5}\cdot\frac{1}{x}=\frac{6(5+\ln x)^{5}}{x}$.

Answer:

$\frac{6(5+\ln x)^{5}}{x}$