Differentiate the following w.r.t. x:
`e^(log(x+sqrt(x^(2)+a^(2))))`

Answer

To differentiate the function \( y = e^{\log(x + \sqrt{x^2 + a^2})} \) with respect to \( x \), we can follow these steps:

### Step 1: Simplify the Expression
We start by simplifying the expression using properties of logarithms and exponentials. Since \( e^{\log(a)} = a \), we can rewrite the function as:
\[ y = x + \sqrt{x^2 + a^2} \]

### Step 2: Differentiate the Function
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x + \sqrt{x^2 + a^2}) \]

### Step 3: Differentiate Each Term
We differentiate each term separately:
1. The derivative of \( x \) is \( 1 \).
2. For the term \( \sqrt{x^2 + a^2} \), we use the chain rule:
\[ \frac{d}{dx}(\sqrt{x^2 + a^2}) = \frac{1}{2\sqrt{x^2 + a^2}} \cdot \frac{d}{dx}(x^2 + a^2) = \frac{1}{2\sqrt{x^2 + a^2}} \cdot (2x) = \frac{x}{\sqrt{x^2 + a^2}} \]

### Step 4: Combine the Derivatives
Now we combine the derivatives:
\[ \frac{dy}{dx} = 1 + \frac{x}{\sqrt{x^2 + a^2}} \]

### Final Result
Thus, the derivative of the given function with respect to \( x \) is:
\[ \frac{dy}{dx} = 1 + \frac{x}{\sqrt{x^2 + a^2}} \]

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