Introduction:
We are given a function, f(x) = x^(2n), and we need to find the nth derivative of this function. The nth derivative represents the rate of change of the function with respect to x, taken n times. To find the nth derivative, we will use the power rule of differentiation repeatedly.

Power Rule of Differentiation:
The power rule states that if we have a function of the form f(x) = x^a, where 'a' is a constant, then the derivative of f(x) with respect to x is given by f'(x) = a * x^(a-1).

Deriving the First Derivative:
To find the first derivative of f(x) = x^(2n), we can apply the power rule:
f'(x) = 2n * x^(2n - 1)

Deriving the Second Derivative:
To find the second derivative, we differentiate f'(x) = 2n * x^(2n - 1) with respect to x using the power rule:
f''(x) = 2n * (2n - 1) * x^(2n - 1 - 1) = (2n)(2n - 1) * x^(2n - 2)

Deriving the nth Derivative:
We can observe a pattern in the derivatives we have derived so far. The nth derivative will have a coefficient that is the product of all the natural numbers from 2 to n, and the exponent of x will be (2n - n) = n. Thus, the nth derivative can be written as:

f^(n)(x) = (2n)(2n - 1)(2n - 2)...(2n - (n - 1)) * x^n

Summary:
- We are given the function f(x) = x^(2n) and asked to find the nth derivative.
- The power rule of differentiation states that the derivative of f(x) = x^a is given by f'(x) = a * x^(a-1).
- Applying the power rule repeatedly, we can find the first, second, and nth derivatives of f(x).
- The nth derivative of f(x) = x^(2n) is given by f^(n)(x) = (2n)(2n - 1)(2n - 2)...(2n - (n - 1)) * x^n.