tan 3x

Answer

To find the differential coefficient (derivative) of the function \( y = \tan(3x) \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function
We have the function:
\[ y = \tan(3x) \]

### Step 2: Apply the chain rule
To differentiate \( y = \tan(3x) \), we will use the chain rule. The derivative of \( \tan(u) \) with respect to \( u \) is \( \sec^2(u) \). Here, \( u = 3x \).

### Step 3: Differentiate the outer function
First, we differentiate the outer function:
\[ \frac{dy}{du} = \sec^2(3x) \]

### Step 4: Differentiate the inner function
Next, we differentiate the inner function \( u = 3x \):
\[ \frac{du}{dx} = 3 \]

### Step 5: Apply the chain rule
Now, we apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \sec^2(3x) \cdot 3 \]

### Step 6: Write the final answer
Thus, the derivative of \( y = \tan(3x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 3 \sec^2(3x) \]

### Summary
The differential coefficient of \( \tan(3x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 3 \sec^2(3x) \]

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