Differentiate `e^(sin^(-1)x)`, w.r.t. x.

Answer

To differentiate the function \( y = e^{\sin^{-1}(x)} \) with respect to \( x \), we will use the chain rule and the derivative of the inverse sine function. Here are the steps:

### Step 1: Identify the function
We have:
\[ y = e^{\sin^{-1}(x)} \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The derivative of \( e^u \) is \( e^u \cdot \frac{du}{dx} \), where \( u = \sin^{-1}(x) \).

So, we have:
\[ \frac{dy}{dx} = e^{\sin^{-1}(x)} \cdot \frac{d}{dx}(\sin^{-1}(x)) \]

### Step 3: Find the derivative of \( \sin^{-1}(x) \)
The derivative of \( \sin^{-1}(x) \) is given by:
\[ \frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1 - x^2}} \]

### Step 4: Substitute back into the derivative
Now, substituting this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = e^{\sin^{-1}(x)} \cdot \frac{1}{\sqrt{1 - x^2}} \]

### Step 5: Final expression
Thus, the final expression for the derivative is:
\[ \frac{dy}{dx} = \frac{e^{\sin^{-1}(x)}}{\sqrt{1 - x^2}} \]

### Summary of the solution
The derivative of \( y = e^{\sin^{-1}(x)} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{e^{\sin^{-1}(x)}}{\sqrt{1 - x^2}} \]