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

Options

A. `8x cos (4x^(2))`
B. `cos (4x^(2))`
C. `-cos(4x^(2))`
D. `-8x cos(4x^(2))`

Explanation

To solve the problem of finding \(\frac{dy_1}{dx}\) where \(y_1 = \sin(4x^2)\), we will use the chain rule of differentiation. Here are the steps:

### Step 1: Identify the function
We have:
\[ y_1 = \sin(4x^2) \]

### Step 2: Use the chain rule
To differentiate \(y_1\) with respect to \(x\), we will apply the chain rule. The chain rule states that if you have a composite function \(y = f(g(x))\), then:
\[ \frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx} \]

In our case, let:
- \(t = 4x^2\)
- Then, \(y_1 = \sin(t)\)

### Step 3: Differentiate \(y_1\) with respect to \(t\)
Now, we differentiate \(y_1\) with respect to \(t\):
\[ \frac{dy_1}{dt} = \cos(t) \]
Substituting back \(t = 4x^2\):
\[ \frac{dy_1}{dt} = \cos(4x^2) \]

### Step 4: Differentiate \(t\) with respect to \(x\)
Next, we differentiate \(t\) with respect to \(x\):
\[ t = 4x^2 \implies \frac{dt}{dx} = 8x \]

### Step 5: Combine results using the chain rule
Now, we can combine these results using the chain rule:
\[ \frac{dy_1}{dx} = \frac{dy_1}{dt} \cdot \frac{dt}{dx} = \cos(4x^2) \cdot 8x \]

### Final Answer
Thus, we have:
\[ \frac{dy_1}{dx} = 8x \cos(4x^2) \]