Differentiate the following functions w.r.t. x.
`f(x) = sqrt( x^2 +1)`

Answer

To differentiate the function \( f(x) = \sqrt{x^2 + 1} \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function
We have:
\[ f(x) = \sqrt{x^2 + 1} \]

### Step 2: Use the chain rule
To differentiate \( f(x) \), we can use the chain rule. The derivative of \( \sqrt{u} \) with respect to \( u \) is:
\[ \frac{d}{du}(\sqrt{u}) = \frac{1}{2\sqrt{u}} \]
where \( u = x^2 + 1 \).

### Step 3: Differentiate the inner function
Now, we need to differentiate the inner function \( u = x^2 + 1 \):
\[ \frac{du}{dx} = \frac{d}{dx}(x^2 + 1) = 2x \]

### Step 4: Apply the chain rule
Now, we can apply the chain rule:
\[ f'(x) = \frac{d}{du}(\sqrt{u}) \cdot \frac{du}{dx} = \frac{1}{2\sqrt{u}} \cdot \frac{du}{dx} \]
Substituting \( u = x^2 + 1 \) and \( \frac{du}{dx} = 2x \):
\[ f'(x) = \frac{1}{2\sqrt{x^2 + 1}} \cdot 2x \]

### Step 5: Simplify the expression
The \( 2 \) in the numerator and denominator cancels out:
\[ f'(x) = \frac{x}{\sqrt{x^2 + 1}} \]

### Final Answer
Thus, the derivative of the function \( f(x) = \sqrt{x^2 + 1} \) with respect to \( x \) is:
\[ f'(x) = \frac{x}{\sqrt{x^2 + 1}} \]

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