The derivative of tan `(x^(@) + 45^(@)) ` , is

Options

A. `(pi)/(180)sec^(2) (x^(@) + 45^(@)) `
B. ` sec^(2)(x^(2) + 45^(@)) `
C. `(180)/(pi)sec^(2) (x^(@) + 45^(@)) `
D. None of these

Explanation

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

### Step 1: Convert degrees to radians
Since the function is in degrees, we need to convert it to radians. The conversion from degrees to radians is done using the formula:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
Thus, we convert \( x + 45^\circ \) to radians:
\[ x + 45^\circ = \left(x + 45\right) \times \frac{\pi}{180} \]

### Step 2: Rewrite the function
Now, we can rewrite the function in terms of radians:
\[ y = \tan\left(\frac{\pi}{180}(x + 45)\right) \]

### Step 3: Differentiate the function
To differentiate \( y \), we use the chain rule. The derivative of \( \tan(u) \) is \( \sec^2(u) \cdot \frac{du}{dx} \), where \( u = \frac{\pi}{180}(x + 45) \).

First, we find \( \frac{du}{dx} \):
\[ u = \frac{\pi}{180}(x + 45) \implies \frac{du}{dx} = \frac{\pi}{180} \]

Now, we can differentiate \( y \):
\[ \frac{dy}{dx} = \sec^2\left(\frac{\pi}{180}(x + 45)\right) \cdot \frac{\pi}{180} \]

### Step 4: Final expression
Thus, the derivative of \( y = \tan(x + 45^\circ) \) is:
\[ \frac{dy}{dx} = \frac{\pi}{180} \sec^2\left(\frac{\pi}{180}(x + 45)\right) \]

### Summary
The derivative of \( \tan(x + 45^\circ) \) is:
\[ \frac{dy}{dx} = \frac{\pi}{180} \sec^2\left(\frac{\pi}{180}(x + 45)\right) \]

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