If a function is written as `:`
`y_(1)=sin(4x^(2)) &` another function is `y_(2)=ln(x^(3))` then `:`
`(dy_(2))/(dx)` will be

Options

A. `(1)/(x^(3))`
B. `(3)/(x)`
C. `-(1)/(x^(3))`
D. `(3)/(x^(2))`

Explanation

To find the derivative of the function \( y_2 = \ln(x^3) \) with respect to \( x \), we can follow these steps:

### Step 1: Identify the function
We have the function:
\[ y_2 = \ln(x^3) \]

### Step 2: Use the property of logarithms
Using the property of logarithms, we can simplify \( \ln(x^3) \):
\[ y_2 = 3 \ln(x) \]

### Step 3: Differentiate the function
Now, we differentiate \( y_2 \) with respect to \( x \):
\[ \frac{dy_2}{dx} = \frac{d}{dx}(3 \ln(x)) \]
Using the constant multiple rule and the derivative of \( \ln(x) \), which is \( \frac{1}{x} \):
\[ \frac{dy_2}{dx} = 3 \cdot \frac{1}{x} \]

### Step 4: Write the final answer
Thus, we can express the derivative as:
\[ \frac{dy_2}{dx} = \frac{3}{x} \]

### Final Answer:
\[ \frac{dy_2}{dx} = \frac{3}{x} \]

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