Differentiate the following w.r.t.x :
`tan(2x+3)`

Answer

To differentiate the function \( y = \tan(2x + 3) \) with respect to \( x \), we will use the chain rule. Here are the steps:

### Step 1: Identify the outer and inner functions
In this case, the outer function is \( \tan(u) \) where \( u = 2x + 3 \).

### Step 2: Differentiate the outer function
The derivative of \( \tan(u) \) with respect to \( u \) is \( \sec^2(u) \).

### Step 3: Differentiate the inner function
Now we differentiate the inner function \( u = 2x + 3 \) with respect to \( x \):
\[ \frac{du}{dx} = 2 \]

### Step 4: Apply the chain rule
Using the chain rule, we have:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \sec^2(2x + 3) \cdot 2 \]

### Step 5: Write the final answer
Thus, the derivative of \( y = \tan(2x + 3) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 2 \sec^2(2x + 3) \]