Determine slope of the tangent at the given value, c. $f(x)=\sqrt{x+1}, a=0$

Step 1 ：Let's find the derivative of the function \(f(x)=\sqrt{x+1}\).

Step 2 ：The derivative of \(f(x)=\sqrt{x+1}\) is \(f'(x)=\frac{1}{2\sqrt{x+1}}\).

Step 3 ：Substitute \(x=0\) into the derivative to find the slope of the tangent line at \(x=0\).

Step 4 ：The slope of the tangent line to the function \(f(x)=\sqrt{x+1}\) at \(x=0\) is \(\boxed{\frac{1}{2}}\).