Introduction:
To find the nth derivative of the function tan^(-1)(x), we can use the general formula for finding the nth derivative of an inverse trigonometric function. In this case, we will use the formula for the derivative of the inverse tangent function.

Formula for the nth derivative of arctan(x):
The formula for finding the nth derivative of the arctan(x) function is given as follows:

d^n/dx^n(arctan(x)) = (-1)^(n-1) * (n-1)! / (1 + x^2)^n

where n is the order of the derivative.

Step-by-step solution:
To find the nth derivative of tan^(-1)(x), we can follow these steps:

1. Apply the formula for the nth derivative of arctan(x):

d^n/dx^n(arctan(x)) = (-1)^(n-1) * (n-1)! / (1 + x^2)^n

2. Simplify the formula:

The formula becomes:

d^n/dx^n(arctan(x)) = (-1)^(n-1) * (n-1)! / (1 + x^2)^n

3. Substitute the value of x with x in the formula:

d^n/dx^n(arctan(x)) = (-1)^(n-1) * (n-1)! / (1 + x^2)^n

4. Simplify the expression further if needed:

If required, you can simplify the expression further based on the specific value of n.

Example:
Let's find the 3rd derivative of tan^(-1)(x) using the formula:

1. Apply the formula for the 3rd derivative:

d^3/dx^3(arctan(x)) = (-1)^(3-1) * (3-1)! / (1 + x^2)^3

2. Simplify the formula:

d^3/dx^3(arctan(x)) = (-1)^2 * 2! / (1 + x^2)^3
= 2 / (1 + x^2)^3

Therefore, the 3rd derivative of tan^(-1)(x) is 2 / (1 + x^2)^3.

Conclusion:
To find the nth derivative of the tan^(-1)(x) function, we can use the formula d^n/dx^n(arctan(x)) = (-1)^(n-1) * (n-1)! / (1 + x^2)^n. By substituting the value of x and simplifying the expression, we can find the specific value of the nth derivative.