If `f(x) = |cosx|`, then `f'(pi/4)` is equal to `"……."`

Explanation

To find \( f'( \frac{\pi}{4} ) \) for the function \( f(x) = |\cos x| \), we will follow these steps:

### Step 1: Determine the value of \( \cos(\frac{\pi}{4}) \)
First, we need to evaluate \( \cos(\frac{\pi}{4}) \):
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

### Step 2: Identify the sign of \( \cos(x) \) in the interval
Since \( \frac{\pi}{4} \) is in the first quadrant, where cosine is positive, we can express \( f(x) \) without the modulus:
\[ f(x) = |\cos x| = \cos x \quad \text{for } x \in \left[0, \frac{\pi}{2}\right] \]

### Step 3: Differentiate \( f(x) \)
Now we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}(\cos x) = -\sin x \]

### Step 4: Evaluate \( f'(\frac{\pi}{4}) \)
Next, we substitute \( x = \frac{\pi}{4} \) into the derivative:
\[ f'\left(\frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) \]

### Step 5: Calculate \( \sin(\frac{\pi}{4}) \)
Now we find \( \sin(\frac{\pi}{4}) \):
\[ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

### Step 6: Final calculation
Putting it all together, we have:
\[ f'\left(\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \]

Thus, the final answer is:
\[ f'\left(\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \]