Differentiate the following functions w.r.t. x:
`tan 2x`

Answer

To differentiate the function \( y = \tan(2x) \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function
Let:
\[ y = \tan(2x) \]

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

### Step 3: Differentiate the outer function
The derivative of \( \tan(u) \) with respect to \( u \) is:
\[ \frac{d}{du}(\tan(u)) = \sec^2(u) \]
Thus, substituting back for \( u \):
\[ \frac{d}{du}(\tan(2x)) = \sec^2(2x) \]

### Step 4: Differentiate the inner function
Next, we differentiate the inner function \( u = 2x \):
\[ \frac{du}{dx} = 2 \]

### Step 5: Combine the derivatives
Now, we can combine the results using the chain rule:
\[ \frac{dy}{dx} = \sec^2(2x) \cdot 2 \]

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

### Final Answer:
\[ \frac{dy}{dx} = 2 \sec^2(2x) \]

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