Step 1
What rule do we use here?
Choose the best differentiation rule.
Step 2
Identify the inner and outer functions
For \( y=(\ln x)^2 \), what are the two functions?
Step 3
Use a substitution
To make the chain rule easier, let
\[
u=\ln x \qquad \text{so} \qquad y=u^2
\]
What is \( \frac{dy}{du} \)?
Step 4
Differentiate the inside function
Now find the derivative of \( \ln x \).
Step 5
How do we combine them?
In the chain rule, we take the derivative of the outside, then ______ by the derivative of the inside.
Step 6
Build the final derivative
Use the chain rule structure, then substitute back in terms of \(x\).
\[
\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}
\]
\[
\frac{dy}{du}=2u
\]
\[
\frac{du}{dx}=\frac{1}{x}
\]
\[
u=\ln x
\]
What is the final derivative \( \frac{dy}{dx} \)?