Understanding the n-th, (n-1)th, and (n-2)nd Derivatives of the Function f(x) x - 1^n

The function f(x) x - 1^n is a mathematical curiosity that can be explored through the lens of calculus, specifically through the process of differentiation. This article delves into understanding how to find the n-th, (n-1)th, and (n-2)nd derivatives of this function using the power rule and other relevant concepts in calculus. By the end, you will grasp the general formula for the k-th derivative of f(x), which is useful for a wide range of applications in mathematics and beyond.

1. Finding the Derivatives of f(x) x - 1^n

To find the n-th, (n-1)th, and (n-2)nd derivatives of the function f(x) x - 1^n, we rely on the power rule for differentiation. The power rule states that if f(x) x^k, then the derivative f'(x) kx^(k-1).

1.1. n-th Derivative

The n-th derivative of the function is calculated by differentiating the function n times. The sequence of alterations due to differentiation reveals a pattern that simplifies the process.

The n-th derivative of f(x) x - 1^n is f^n(x) n!. This result is derived from the fact that each differentiation reduces the power of x - 1 by 1, and the coefficients multiply by n, n-1, n-2, and so on, until reaching 1. When the power of x - 1 reaches 0, the term vanishes, leaving only the factorial of n.

The final result can be summarized as:

[f^n(x) n!]

1.2. (n-1)th Derivative

The (n-1)th derivative of f(x) x - 1^n can be found by differentiating the function n-1 times.

We start with (f^{n-1}(x) n cdot (x - 1)^{n-1}). Continuing to differentiate, the derivative is simplified to (n cdot n-1 cdot (x - 1)^{n-2}). This pattern continues, resulting in (f^{n-1}(x) n cdot (n-1) cdot (x - 1)^{n-1}).

The n-1th derivative can be succinctly written as:

[f^{n-1}(x) n cdot (x - 1)^{(n-1)}]

1.3. (n-2)nd Derivative

Similarly, the (n-2)nd derivative can be found by differentiating the (n-1)th derivative one more time.

Starting from (f^{n-1}(x) n cdot (x - 1)^{n-1}), the (n-2)nd derivative is: [f^{n-2}(x) n cdot (n-1) cdot (x - 1)^{n-2}] Generalizing, the (n-2)nd derivative follows the pattern: [f^{n-2}(x) n cdot (n-1) cdot (x - 1)^{n-2}]

The (n-2)nd derivative can be expressed in a general form as:

[f^{n-2}(x) n cdot (n-1) cdot (x - 1)^{n-2}]

2. Summary of Derivatives

Summarizing, the derivatives are:

n-th Derivative:[f^n(x) n!] (n-1)th Derivative:[f^{n-1}(x) n cdot (x - 1)^{n-1}] (n-2)nd Derivative:[f^{n-2}(x) n cdot (n-1) cdot (x - 1)^{n-2}]

3. Generalizing the Derivatives

For any natural number (k) such that 0 leq k leq n, the k-th derivative of f(x) x - 1^n can be generalized as:

[f^k(x) n(n-1)(n-2) cdots (n-k 1)(x - 1)^{n-k}]

This general formula encapsulates the process of differentiation and is a powerful tool for understanding the behavior of functions under repeated differentiation.

4. Practical Application and Verification

To verify that this general formula holds true, let's consider an example where n 3:

[y (x - 1)^3] The first derivative is y' 3(x - 1)^2 The second derivative is y'' 3 cdot 2(x - 1) 6(x - 1) The third derivative is y''' 3 cdot 2 cdot 1 6

This example confirms that the k-th derivative can be calculated as:

[y^{(k)} 3 cdot 2 cdot 1 cdots (4 - k) (x - 1)^{3-k}]

For any (k), the factorial of the initial coefficient ensures the correct pattern of differentiation.

5. Conclusion

In conclusion, understanding the n-th, (n-1)th, and (n-2)nd derivatives of the function f(x) x - 1^n provides valuable insights into the power rule and factorial functions in calculus. This knowledge is not only theoretical but also practical, with applications in various fields such as physics, engineering, and economics.