`e^(cotx)`

Answer

To differentiate the function \( y = e^{\cot x} \), we will use the chain rule. Here’s the step-by-step solution:

### Step 1: Identify the function
We have the function:
\[ y = e^{\cot x} \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The chain rule states that if you have a function \( y = e^{u} \), where \( u = \cot x \), then:
\[ \frac{dy}{dx} = e^{u} \cdot \frac{du}{dx} \]
In our case, \( u = \cot x \).

### Step 3: Differentiate \( u = \cot x \)
Now we need to find \( \frac{du}{dx} \):
\[ \frac{du}{dx} = \frac{d}{dx}(\cot x) = -\csc^2 x \]

### Step 4: Substitute back into the chain rule
Now, substituting \( u \) and \( \frac{du}{dx} \) back into the chain rule:
\[ \frac{dy}{dx} = e^{\cot x} \cdot (-\csc^2 x) \]

### Step 5: Simplify the expression
Thus, we can write the final answer as:
\[ \frac{dy}{dx} = -e^{\cot x} \csc^2 x \]

### Final Answer
The derivative of \( y = e^{\cot x} \) is:
\[ \frac{dy}{dx} = -e^{\cot x} \csc^2 x \]

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