Is the question sqrt(1+x^2)? If so, then this can be written (1+x^2)^(1/2). The square root is changed to the power of a half. To get the derivative we have to treat (1+x^2) as if it were just x, so we need to find the derivative as if it were x^(1/2). That gives us (1/2)(1+x^2)^((1/2)-1) which comes to (1/2)(1+x^2)^(-1/2). Then we need to get the derivative of (1+x^2) itself, which is 2x and multiply our partial result by this, to give 2x(1/2)(1+x^2)^(-1/2). So the final result can be written several different ways x/sqrt(1+x^2) or x/(1+x^2)^(1/2) or x(1+x^2)^(-1/2).
The second derivative is obtained by splitting the first derivative into two parts: x and (1+x^2)^(-1/2). We use the rule "hold one and multiply by the derivative of the other and add this to the derivative of the one multiplied by the other". So we hold x and multiply by the derivative of (1+x^2)^(-1/2), which is (-1/2)(1+x^2)^(-3/2)*2x, i.e., -x(1+x^2)^(-3/2). So the first part of the answer is x times this, i.e., -x^2(1+x^2)^(-3/2) and the second part of the answer is easier, it's just (1+x^2)^(-1/2) times the derivative of x, which is just 1. So the second derivative becomes the two parts added together: -x^2(1+x^2)^(-3/2) + (1+x^2)^(-1/2). It can be simplified a little: (1+x^2)^(-1/2)(-x^2/(1+x^2)+1). This further simplifies: (1+x^2)^(-1/2)(1+x^2-x^2)/(1+x^2) = (1+x^2)^(-3/2), because two x^2 terms cancel out in the brackets. So the second derivative is simply (1+x^2)^(-3/2).
The third derivative, following the rules above, is -3/2(1+x^2)^(-5/2) * 2x = -3x(1+x^2)^(-5/2).