derivative of tan 3x

Answer

To find the derivative of the function \( f(x) = \tan(3x) \), we can follow these steps:

### Step 1: Identify the function and its derivative
The given function is \( f(x) = \tan(3x) \). We know that the derivative of \( \tan(x) \) is \( \sec^2(x) \).

**Hint:** Recall the basic derivative rule for the tangent function.

### Step 2: Apply the chain rule
Since \( 3x \) is a function of \( x \), we need to apply the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by \( f'(g(x)) \cdot g'(x) \).

**Hint:** Remember that the chain rule is used when differentiating composite functions.

### Step 3: Differentiate the outer function
Using the chain rule, we differentiate the outer function \( \tan(u) \) where \( u = 3x \):
\[ \frac{d}{dx}(\tan(3x)) = \sec^2(3x) \cdot \frac{d}{dx}(3x) \]

**Hint:** Identify the outer function and its derivative.

### Step 4: Differentiate the inner function
Now we need to differentiate the inner function \( 3x \):
\[ \frac{d}{dx}(3x) = 3 \]

**Hint:** The derivative of a linear function \( kx \) is simply \( k \).

### Step 5: Combine the results
Now we can combine the results from Steps 3 and 4:
\[ \frac{d}{dx}(\tan(3x)) = \sec^2(3x) \cdot 3 \]

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

**Final Answer:** The derivative of \( \tan(3x) \) is \( 3 \sec^2(3x) \).

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