Find the derivative of `y = ln x^2`

Answer

To find the derivative of the function \( y = \ln(x^2) \), we can follow these steps:

### Step 1: Simplify the Function
We can use the properties of logarithms to simplify the function before differentiating. Recall that \( \ln(a^b) = b \ln(a) \). Therefore, we can rewrite the function as:
\[ y = \ln(x^2) = 2 \ln(x) \]

### Step 2: Differentiate the Simplified Function
Now we can differentiate \( y = 2 \ln(x) \) with respect to \( x \). The derivative of \( \ln(x) \) is \( \frac{1}{x} \). Thus, we have:
\[ \frac{dy}{dx} = 2 \cdot \frac{d}{dx}[\ln(x)] = 2 \cdot \frac{1}{x} \]

### Step 3: Write the Final Derivative
Putting it all together, we find:
\[ \frac{dy}{dx} = \frac{2}{x} \]

### Final Answer
The derivative of \( y = \ln(x^2) \) is:
\[ \frac{dy}{dx} = \frac{2}{x} \]

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