Let `y=tan((pi)/4+x/2)`. Then `(dy)/(dx)` at `x=(pi)/4`

Options

A. is 1
B. is -1
C. does not exist
D. None of these

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \tan\left(\frac{\pi}{4} + \frac{x}{2}\right)\) at \(x = \frac{\pi}{4}\), we will follow these steps:

### Step 1: Differentiate \(y\) with respect to \(x\)

We start with the function:
\[ y = \tan\left(\frac{\pi}{4} + \frac{x}{2}\right) \]

Using the chain rule, the derivative \(\frac{dy}{dx}\) can be expressed as:
\[ \frac{dy}{dx} = \sec^2\left(\frac{\pi}{4} + \frac{x}{2}\right) \cdot \frac{d}{dx}\left(\frac{\pi}{4} + \frac{x}{2}\right) \]

### Step 2: Differentiate the inner function

Now we differentiate the inner function:
\[ \frac{d}{dx}\left(\frac{\pi}{4} + \frac{x}{2}\right) = 0 + \frac{1}{2} = \frac{1}{2} \]

### Step 3: Substitute back into the derivative

Now we can substitute this back into our expression for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \sec^2\left(\frac{\pi}{4} + \frac{x}{2}\right) \cdot \frac{1}{2} \]

### Step 4: Evaluate \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\)

Next, we evaluate \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\):
\[ \frac{dy}{dx}\bigg|_{x=\frac{\pi}{4}} = \frac{1}{2} \sec^2\left(\frac{\pi}{4} + \frac{\frac{\pi}{4}}{2}\right) \]

Calculating the argument:
\[ \frac{\pi}{4} + \frac{\frac{\pi}{4}}{2} = \frac{\pi}{4} + \frac{\pi}{8} = \frac{2\pi}{8} + \frac{\pi}{8} = \frac{3\pi}{8} \]

So we have:
\[ \frac{dy}{dx}\bigg|_{x=\frac{\pi}{4}} = \frac{1}{2} \sec^2\left(\frac{3\pi}{8}\right) \]

### Final Answer

Thus, the value of \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\) is:
\[ \frac{1}{2} \sec^2\left(\frac{3\pi}{8}\right) \]

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