To find the derivative of the function \( y = e^x \cot x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u \) and \( v \), then the derivative of their product is given by:
\[
\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}
\]
### Step 1: Identify the functions
Let:
- \( u = e^x \)
- \( v = \cot x \)
### Step 2: Differentiate \( u \) and \( v \)
Now, we need to find the derivatives of \( u \) and \( v \):
- The derivative of \( u \) is:
\[
\frac{du}{dx} = e^x
\]
- The derivative of \( v \) is:
\[
\frac{dv}{dx} = -\csc^2 x
\]
### Step 3: Apply the product rule
Now we apply the product rule:
\[
\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}
\]
Substituting the values we found:
\[
\frac{dy}{dx} = e^x \cdot (-\csc^2 x) + \cot x \cdot e^x
\]
### Step 4: Simplify the expression
We can factor out \( e^x \):
\[
\frac{dy}{dx} = e^x (-\csc^2 x + \cot x)
\]
This can be rewritten as:
\[
\frac{dy}{dx} = e^x (\cot x - \csc^2 x)
\]
### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[
\frac{dy}{dx} = e^x (\cot x - \csc^2 x)
\]
To find the derivative of the function \( y = e^x \cot x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u \) and \( v \), then the derivative of their product is given by:
\[
\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}
\]
### Step 1: Identify the functions
Let:
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