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 \( \cos(\frac{\pi}{4}) \):
\[ \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} > 0 \]
Since \( \cos(\frac{\pi}{4}) \) is positive, we can conclude that \( |\cos x| = \cos x \) in the interval around \( \frac{\pi}{4} \).

### Step 2: Rewrite the function
Since \( \cos x \) is positive at \( x = \frac{\pi}{4} \), we can write:
\[ f(x) = \cos x \]

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

### Step 4: Evaluate the derivative at \( x = \frac{\pi}{4} \)
Now we substitute \( x = \frac{\pi}{4} \) into the derivative:
\[ f'\left(\frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) \]
We know that:
\[ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
Thus,
\[ f'\left(\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}} \]

### Final Answer
Therefore, \( f'(\frac{\pi}{4}) = -\frac{1}{\sqrt{2}} \).

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