Differentiate the following function w.r.t. x:
`cot nx`

Answer

To differentiate the function \( y = \cot(nx) \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function
Let \( y = \cot(nx) \).

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we will use the chain rule. The derivative of \( \cot(\theta) \) with respect to \( \theta \) is \( -\csc^2(\theta) \). Here, \( \theta = nx \).

Thus, we have:
\[ \frac{dy}{dx} = \frac{d}{dx}(\cot(nx)) = -\csc^2(nx) \cdot \frac{d}{dx}(nx) \]

### Step 3: Differentiate \( nx \)
Now, we differentiate \( nx \) with respect to \( x \):
\[ \frac{d}{dx}(nx) = n \]

### Step 4: Substitute back into the derivative
Now we substitute back into our derivative expression:
\[ \frac{dy}{dx} = -\csc^2(nx) \cdot n \]

### Step 5: Final result
Thus, the derivative of \( y = \cot(nx) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -n \csc^2(nx) \]

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