Find the derivative of `y = ln x^4`

Answer

To find the derivative of the function \( y = \ln(x^4) \), we can follow these steps:

### Step 1: Apply the logarithmic property
Using the property of logarithms that states \( \ln(a^b) = b \ln(a) \), we can rewrite the function:
\[ y = \ln(x^4) = 4 \ln(x) \]

### Step 2: Differentiate the function
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(4 \ln(x)) \]
Using the constant multiple rule, we can factor out the constant:
\[ \frac{dy}{dx} = 4 \cdot \frac{d}{dx}(\ln(x)) \]

### Step 3: Differentiate \( \ln(x) \)
The derivative of \( \ln(x) \) is \( \frac{1}{x} \):
\[ \frac{dy}{dx} = 4 \cdot \frac{1}{x} \]

### Step 4: Simplify the result
Thus, we can simplify the expression:
\[ \frac{dy}{dx} = \frac{4}{x} \]

### Final Answer
The derivative of \( y = \ln(x^4) \) is:
\[ \frac{dy}{dx} = \frac{4}{x} \]

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