find the derivatives of the following :
`cos 3x`

Answer

To find the derivative of the function \( y = \cos(3x) \), we will use the chain rule of differentiation. Here’s how to do it step by step:

### Step 1: Identify the outer and inner functions
In the function \( y = \cos(3x) \), we can identify:
- Outer function: \( u = \cos(x) \)
- Inner function: \( v = 3x \)

### Step 2: Differentiate the outer function
The derivative of the outer function \( u = \cos(v) \) with respect to \( v \) is:
\[ \frac{du}{dv} = -\sin(v) \]

### Step 3: Differentiate the inner function
The derivative of the inner function \( v = 3x \) with respect to \( x \) is:
\[ \frac{dv}{dx} = 3 \]

### Step 4: Apply the chain rule
Using the chain rule, we find the derivative of \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = -\sin(3x) \cdot 3 \]

### Step 5: Simplify the expression
Thus, we can express the derivative as:
\[ \frac{dy}{dx} = -3\sin(3x) \]

### Final Answer
The derivative of \( \cos(3x) \) is:
\[ \frac{dy}{dx} = -3\sin(3x) \]

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