Differentiate the following functions w.r.t. x:
`sec ((x)/(2) -1)`

Answer

To differentiate the function \( y = \sec\left(\frac{x}{2} - 1\right) \) with respect to \( x \), we can follow these steps:

### Step 1: Identify the function
Let:
\[ y = \sec\left(\frac{x}{2} - 1\right) \]

### Step 2: Use the chain rule
To differentiate \( y \) with respect to \( x \), we will use the chain rule. First, we set:
\[ u = \frac{x}{2} - 1 \]
Then, we can rewrite \( y \) in terms of \( u \):
\[ y = \sec(u) \]

### Step 3: Differentiate \( y \) with respect to \( u \)
The derivative of \( \sec(u) \) with respect to \( u \) is:
\[ \frac{dy}{du} = \sec(u) \tan(u) \]

### Step 4: Differentiate \( u \) with respect to \( x \)
Now, we differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{1}{2} \]

### Step 5: Apply the chain rule
Using the chain rule, we can find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \sec(u) \tan(u) \cdot \frac{1}{2} \]

### Step 6: Substitute back for \( u \)
Now we substitute back \( u = \frac{x}{2} - 1 \):
\[ \frac{dy}{dx} = \sec\left(\frac{x}{2} - 1\right) \tan\left(\frac{x}{2} - 1\right) \cdot \frac{1}{2} \]

### Final Answer
Thus, the derivative of \( y = \sec\left(\frac{x}{2} - 1\right) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{2} \sec\left(\frac{x}{2} - 1\right) \tan\left(\frac{x}{2} - 1\right) \]

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