If `y=sec x^(@)`, then `(dy)/(dx)=`

Options

A. sec x tan x
B. `sec x^(@) tan x^(@)`
C. `pi/180 sec x^(@) tan x^(@)`
D. `180/pi sec x^(@) tan x^(@)`

Answer

To find the derivative of \( y = \sec x^\circ \) with respect to \( x \), we will follow these steps:

### Step 1: Understand the function
We have \( y = \sec x^\circ \). The secant function is defined as:
\[ \sec x = \frac{1}{\cos x} \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we will use the chain rule. The derivative of \( \sec u \) with respect to \( u \) is \( \sec u \tan u \). Here, \( u = x^\circ \).

### Step 3: Convert degrees to radians
Since the derivative is usually calculated in radians, we need to convert degrees to radians. The conversion from degrees to radians is given by:
\[ x^\circ = x \cdot \frac{\pi}{180} \]

### Step 4: Apply the chain rule
Using the chain rule, we have:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
where \( u = x^\circ \).

### Step 5: Differentiate \( y \) with respect to \( u \)
Now, we differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = \sec u \tan u \]

### Step 6: Differentiate \( u \) with respect to \( x \)
Next, we differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}\left(x \cdot \frac{\pi}{180}\right) = \frac{\pi}{180} \]

### Step 7: Combine the results
Now, we can combine the results:
\[ \frac{dy}{dx} = \sec(x^\circ) \tan(x^\circ) \cdot \frac{\pi}{180} \]

### Final Result
Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{\pi}{180} \sec(x^\circ) \tan(x^\circ) \]