` If y=cos (2x +45^(@) ),then (dy)/(dx) =`

Options

A. ` 2sin (2x +45^(@) ) `
B. ` -2sin (2x +45^(@) ) `
C. ` (pi)/( 90) sin (2x +45^(@) ) `
D. ` (-pi)/( 90 ) sin ( 2x + 45^(@) ) `

Answer

To find the derivative of the function \( y = \cos(2x + 45^\circ) \), we will follow these steps:

### Step 1: Identify the function
We start with the function:
\[ y = \cos(2x + 45^\circ) \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we will use the chain rule. The derivative of \( \cos(u) \) is \( -\sin(u) \cdot \frac{du}{dx} \), where \( u = 2x + 45^\circ \).

### Step 3: Find \( \frac{du}{dx} \)
First, we differentiate \( u \):
\[ u = 2x + 45^\circ \]
The derivative \( \frac{du}{dx} \) is:
\[ \frac{du}{dx} = 2 \]

### Step 4: Apply the chain rule
Now, we apply the chain rule:
\[ \frac{dy}{dx} = -\sin(u) \cdot \frac{du}{dx} \]
Substituting \( u \) and \( \frac{du}{dx} \):
\[ \frac{dy}{dx} = -\sin(2x + 45^\circ) \cdot 2 \]

### Step 5: Simplify the expression
Thus, we can simplify this to:
\[ \frac{dy}{dx} = -2\sin(2x + 45^\circ) \]

### Final Answer
The derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -2\sin(2x + 45^\circ) \]

---