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 expression for \( f(x) \)

Since \( f(x) = |\cos x| \), we need to analyze when \( \cos x \) is positive or negative.

### Step 2: Analyze \( \cos x \) at \( x = \frac{\pi}{4} \)

At \( x = \frac{\pi}{4} \), we calculate:
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} > 0 \]
Since \( \cos\left(\frac{\pi}{4}\right) \) is positive, we can write:
\[ f(x) = \cos x \quad \text{for } x = \frac{\pi}{4} \]

### Step 3: Differentiate \( f(x) \)

Since \( f(x) = \cos x \) in the neighborhood of \( x = \frac{\pi}{4} \), we can differentiate \( f(x) \):
\[ f'(x) = -\sin x \]

### Step 4: Evaluate \( f'(\frac{\pi}{4}) \)

Now, we substitute \( x = \frac{\pi}{4} \) into the derivative:
\[ f'\left(\frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) \]
Calculating \( \sin\left(\frac{\pi}{4}\right) \):
\[ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
Thus,
\[ f'\left(\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \]

### Final Answer
\[ f'\left(\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \]

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