Find the derivative of the following w.r.t. x :
`cos^(-1)(e^(x))`

Answer

To find the derivative of \( y = \cos^{-1}(e^x) \) with respect to \( x \), we will use the chain rule and the derivative of the inverse cosine function.

### Step-by-Step Solution:

1. **Define the function**:
Let \( y = \cos^{-1}(e^x) \).

2. **Differentiate using the chain rule**:
The derivative of \( \cos^{-1}(u) \) with respect to \( u \) is given by:
\[ \frac{d}{du}(\cos^{-1}(u)) = -\frac{1}{\sqrt{1 - u^2}} \]
Here, \( u = e^x \). Therefore, we need to apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

3. **Find \( \frac{du}{dx} \)**:
Since \( u = e^x \), we differentiate:
\[ \frac{du}{dx} = e^x \]

4. **Substitute into the chain rule**:
Now substituting \( u = e^x \) into the derivative of \( \cos^{-1}(u) \):
\[ \frac{dy}{dx} = -\frac{1}{\sqrt{1 - (e^x)^2}} \cdot e^x \]

5. **Simplify the expression**:
This gives us:
\[ \frac{dy}{dx} = -\frac{e^x}{\sqrt{1 - e^{2x}}} \]

### Final Answer:
Thus, the derivative of \( y = \cos^{-1}(e^x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{e^x}{\sqrt{1 - e^{2x}}} \]