If `y=[sin""(x)/(2)+cos ""(x)/(2)]^(2)`,find`(dy)/(dx)"at"x (pi)/(6)`

Options

A. `(1)/(2)`
B. `(sqrt(3))/(2)`
C. 1
D. 0

Answer

To find the derivative \(\frac{dy}{dx}\) of the function \(y = \left(\sin\left(\frac{x}{2}\right) + \cos\left(\frac{x}{2}\right)\right)^2\) at \(x = \frac{\pi}{6}\), we will follow these steps:

### Step 1: Expand the function
We start with the given function:
\[ y = \left(\sin\left(\frac{x}{2}\right) + \cos\left(\frac{x}{2}\right)\right)^2 \]
Using the formula for the square of a sum, \( (a + b)^2 = a^2 + b^2 + 2ab \), we can expand this:
\[ y = \sin^2\left(\frac{x}{2}\right) + \cos^2\left(\frac{x}{2}\right) + 2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right) \]

### Step 2: Simplify using trigonometric identities
Using the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\) and the double angle identity \(2\sin\theta\cos\theta = \sin(2\theta)\), we can simplify:
\[ y = 1 + \sin(x) \]

### Step 3: Differentiate the function
Now, we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = 0 + \cos(x) = \cos(x) \]

### Step 4: Evaluate the derivative at \(x = \frac{\pi}{6}\)
Now we need to find \(\frac{dy}{dx}\) at \(x = \frac{\pi}{6}\):
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{6}} = \cos\left(\frac{\pi}{6}\right) \]
We know that \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\).

### Final Result
Thus, the value of \(\frac{dy}{dx}\) at \(x = \frac{\pi}{6}\) is:
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{6}} = \frac{\sqrt{3}}{2} \]

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