If `y=|tanx-|sinx||`, then the value of `(dy)/(dx)` at `x=(5pi)/(4)` is

Options

A. `(2sqrt2+1)/(sqrt2)`
B. `(2sqrt2-1)/(sqrt2)`
C. `(sqrt2+1)/(2)`
D. `(sqrt2-1)/(2)`

Answer

To find the value of \(\frac{dy}{dx}\) at \(x = \frac{5\pi}{4}\) for the function \(y = |\tan x - |\sin x||\), we will follow these steps:

### Step 1: Determine the values of \(\tan x\) and \(\sin x\) at \(x = \frac{5\pi}{4}\)

At \(x = \frac{5\pi}{4}\):
- \(\tan\left(\frac{5\pi}{4}\right) = \tan\left(\pi + \frac{\pi}{4}\right) = \tan\left(\frac{\pi}{4}\right) = 1\)
- \(\sin\left(\frac{5\pi}{4}\right) = \sin\left(\pi + \frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}}\)

### Step 2: Evaluate the expression for \(y\)

Now substituting these values into the expression for \(y\):
\[ y = |\tan\left(\frac{5\pi}{4}\right) - |\sin\left(\frac{5\pi}{4}\right)|| = |1 - |-\frac{1}{\sqrt{2}}|| = |1 - \frac{1}{\sqrt{2}}| \]

Since \(\frac{1}{\sqrt{2}} \approx 0.707\), we have:
\[ y = |1 - 0.707| = |0.293| = 0.293 \]

### Step 3: Determine the expression for \(y\) in the relevant interval

Since \(\tan x\) is positive and \(\sin x\) is negative in the third quadrant where \(x = \frac{5\pi}{4}\), we can express \(y\) as:
\[ y = \tan x + |\sin x| = \tan x + (-\sin x) = \tan x + \sin x \]

### Step 4: Differentiate \(y\)

Now we differentiate \(y\):
\[ \frac{dy}{dx} = \frac{d}{dx}(\tan x + \sin x) = \sec^2 x + \cos x \]

### Step 5: Evaluate \(\frac{dy}{dx}\) at \(x = \frac{5\pi}{4}\)

Now we substitute \(x = \frac{5\pi}{4}\) into the derivative:
\[ \sec^2\left(\frac{5\pi}{4}\right) = \sec^2\left(\pi + \frac{\pi}{4}\right) = \sec^2\left(\frac{\pi}{4}\right) = 2 \]
\[ \cos\left(\frac{5\pi}{4}\right) = -\frac{1}{\sqrt{2}} \]

Thus:
\[ \frac{dy}{dx} = 2 + \left(-\frac{1}{\sqrt{2}}\right) = 2 - \frac{1}{\sqrt{2}} \]

### Final Answer

The value of \(\frac{dy}{dx}\) at \(x = \frac{5\pi}{4}\) is:
\[ \frac{dy}{dx} = 2 - \frac{1}{\sqrt{2}} \]

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