`sin 4x`

Answer

To differentiate the function \( y = \sin(4x) \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function
Let:
\[ y = \sin(4x) \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
In our case, \( f(u) = \sin(u) \) where \( u = 4x \).

### Step 3: Differentiate the outer function
The derivative of \( \sin(u) \) with respect to \( u \) is:
\[ \frac{d}{du}(\sin(u)) = \cos(u) \]
Substituting back \( u = 4x \):
\[ \frac{d}{du}(\sin(4x)) = \cos(4x) \]

### Step 4: Differentiate the inner function
Next, we differentiate the inner function \( u = 4x \):
\[ \frac{du}{dx} = 4 \]

### Step 5: Apply the chain rule
Now, we combine these results using the chain rule:
\[ \frac{dy}{dx} = \cos(4x) \cdot 4 \]

### Step 6: Final result
Thus, the derivative of \( y = \sin(4x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 4 \cos(4x) \]