derivative of `sinx^@`

Answer

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

### Step 1: Convert Degrees to Radians
We start with the function \( \sin(x^\circ) \). To differentiate this function, we need to convert the angle from degrees to radians.

We know that:
\[ 180^\circ = \pi \text{ radians} \]
Thus, \( 1^\circ = \frac{\pi}{180} \text{ radians} \).

So, we can express \( x^\circ \) in radians as:
\[ x^\circ = x \cdot \frac{\pi}{180} \]

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

### Step 3: Differentiate the Function
Next, we will differentiate \( \sin\left(\frac{\pi x}{180}\right) \) using the chain rule. The derivative of \( \sin(u) \) is \( \cos(u) \) where \( u = \frac{\pi x}{180} \).

Using the chain rule:
\[ \frac{d}{dx} \sin\left(\frac{\pi x}{180}\right) = \cos\left(\frac{\pi x}{180}\right) \cdot \frac{d}{dx}\left(\frac{\pi x}{180}\right) \]

### Step 4: Find the Inner Derivative
Now we need to find the derivative of 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:
\[ \frac{d}{dx} \sin\left(\frac{\pi x}{180}\right) = \cos\left(\frac{\pi x}{180}\right) \cdot \frac{\pi}{180} \]

Thus, the derivative of \( \sin(x^\circ) \) is:
\[ \frac{\pi}{180} \cos\left(\frac{\pi x}{180}\right) \]

### Final Result
The derivative of \( \sin(x^\circ) \) is:
\[ \frac{d}{dx} \sin(x^\circ) = \frac{\pi}{180} \cos\left(\frac{\pi x}{180}\right) \]

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