derivative of `Cosx^@`

Answer

To find the derivative of \( \cos(x^\circ) \), we will follow these steps:

### Step 1: Convert degrees to radians
We know that \( 180^\circ \) is equivalent to \( \pi \) radians. Therefore, to convert \( x^\circ \) to radians, we use the conversion factor:
\[ x^\circ = \frac{\pi x}{180} \]

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

### Step 3: Differentiate the function
Next, we will differentiate \( \cos\left(\frac{\pi x}{180}\right) \) using the chain rule. The derivative of \( \cos(u) \) is \( -\sin(u) \), where \( u = \frac{\pi x}{180} \). Thus, we have:
\[ \frac{d}{dx} \cos\left(\frac{\pi x}{180}\right) = -\sin\left(\frac{\pi x}{180}\right) \cdot \frac{d}{dx}\left(\frac{\pi x}{180}\right) \]

### Step 4: Find the derivative of the inner function
Now we need to differentiate the inner function \( \frac{\pi x}{180} \):
\[ \frac{d}{dx}\left(\frac{\pi x}{180}\right) = \frac{\pi}{180} \]

### Step 5: Combine the results
Now we can combine the results from Step 3 and Step 4:
\[ \frac{d}{dx} \cos(x^\circ) = -\sin\left(\frac{\pi x}{180}\right) \cdot \frac{\pi}{180} \]

### Final Answer
Thus, the derivative of \( \cos(x^\circ) \) is:
\[ \frac{d}{dx} \cos(x^\circ) = -\frac{\pi}{180} \sin\left(\frac{\pi x}{180}\right) \]

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