Question

1. attempt 1: 10 attempts remaining. let ( f(x) = (ln x)^3 ) ( f(x) = ) submit answer next item practice similar

Explanation:

Step1: Identify the function type

The function \( f(x) = (\ln x)^3 \) is a composite function. Let \( u = \ln x \), so \( f(u)=u^3 \).

Step2: Apply the chain rule

The chain rule states that \( f'(x)=\frac{df}{du}\cdot\frac{du}{dx} \). First, find the derivative of \( f(u) \) with respect to \( u \): \( \frac{df}{du}=3u^2 \) (using the power rule \( \frac{d}{du}(u^n)=nu^{n - 1} \) with \( n = 3 \)). Then, find the derivative of \( u=\ln x \) with respect to \( x \): \( \frac{du}{dx}=\frac{1}{x} \) (since the derivative of \( \ln x \) is \( \frac{1}{x} \)).

Step3: Substitute back \( u=\ln x \)

Substitute \( u = \ln x \) into \( \frac{df}{du} \) and multiply by \( \frac{du}{dx} \): \( f'(x)=3(\ln x)^2\cdot\frac{1}{x}=\frac{3(\ln x)^2}{x} \).

Answer:

\( \frac{3(\ln x)^2}{x} \)