If `y=|tan (pi/4-x)|`, then `(dy)/(dx)` at `x=pi/4` is

Options

A. `-1`
B. 1
C. does not exist
D. none

Answer

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

### Step 1: Define the Function
We start with the function:
\[ y = |\tan\left(\frac{\pi}{4} - x\right)| \]

### Step 2: Analyze the Function
We need to determine the behavior of \(\tan\left(\frac{\pi}{4} - x\right)\) around \(x = \frac{\pi}{4}\):
- When \(x < \frac{\pi}{4}\), \(\frac{\pi}{4} - x > 0\) and thus \(\tan\left(\frac{\pi}{4} - x\right) > 0\).
- When \(x = \frac{\pi}{4}\), \(\tan(0) = 0\).
- When \(x > \frac{\pi}{4}\), \(\frac{\pi}{4} - x < 0\) and thus \(\tan\left(\frac{\pi}{4} - x\right) < 0\).

Hence, we can express \(y\) as:
\[ y = \begin{cases} \tan\left(\frac{\pi}{4} - x\right) & \text{if } x < \frac{\pi}{4} \\ 0 & \text{if } x = \frac{\pi}{4} \\ -\tan\left(\frac{\pi}{4} - x\right) & \text{if } x > \frac{\pi}{4} \end{cases} \]

### Step 3: Differentiate the Function
Now we differentiate \(y\) with respect to \(x\) in the three cases:

1. **For \(x < \frac{\pi}{4}\)**:
\[ \frac{dy}{dx} = \sec^2\left(\frac{\pi}{4} - x\right) \cdot (-1) = -\sec^2\left(\frac{\pi}{4} - x\right) \]

2. **For \(x = \frac{\pi}{4}\)**:
\[ \frac{dy}{dx} = 0 \quad \text{(as } y = 0 \text{ at this point)} \]

3. **For \(x > \frac{\pi}{4}\)**:
\[ \frac{dy}{dx} = -\sec^2\left(\frac{\pi}{4} - x\right) \cdot (-1) = \sec^2\left(\frac{\pi}{4} - x\right) \]

### Step 4: Evaluate the Derivatives at \(x = \frac{\pi}{4}\)
Next, we need to evaluate the left-hand derivative and right-hand derivative at \(x = \frac{\pi}{4}\):

- **Left-hand derivative**:
\[ \lim_{h \to 0^-} \frac{dy}{dx} = -\sec^2(0) = -1 \]

- **Right-hand derivative**:
\[ \lim_{h \to 0^+} \frac{dy}{dx} = \sec^2(0) = 1 \]

### Step 5: Conclusion
Since the left-hand derivative \(-1\) does not equal the right-hand derivative \(1\), the derivative \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\) does not exist.

Thus, the final answer is:
\[ \frac{dy}{dx} \text{ at } x = \frac{\pi}{4} \text{ does not exist.} \]