If f(x)=x^-n for n in 𝐍, prove that f^(k)(x) =(-1)^k...

If $f(x)=x^{-n}$ for $n$ in $\mathbf{N},$ prove that
$$
\begin{aligned}
f^{(k)}(x) &=(-1)^{k} \frac{(n+k-1) !}{(k-1) !} x^{-n-k} \\
&=(-1)^{k} n !\left(\begin{array}{c}
n+k-1 \\
k-1
\end{array}\right) x^{-n-k}, \text { for } x \neq 0
\end{aligned}
$$

Key Concepts

Power Rule for Differentiation

This concept refers to the fundamental rule in calculus that allows the differentiation of a function of the form x^m, where m is a constant. The rule states that the derivative of x^m is m*x^(m-1), and it is the basic tool used to differentiate functions with power expressions, including negative exponents. In higher order derivatives, the power rule is applied repeatedly, and the emerging pattern forms the basis of the general formula.

Iterative Differentiation

Iterative differentiation involves applying the derivative process multiple times to obtain higher order derivatives. In this context, each derivative introduces additional factors that can be expressed in terms of factorials and combinatorial coefficients. Recognizing the pattern in successive derivatives is crucial to establishing a general formula for the k-th derivative.

Factorial Notation

Factorial notation (n!) is a product of all positive integers up to n and commonly appears in formulas involving series, combinatorics, and higher order differentiation. In this problem, factorials arise from the product of decreasing linear factors during repeated application of the power rule, enabling the compact formulation of the derivative in terms of n and k.

Binomial Coefficients

Binomial coefficients, usually expressed as 'n choose k', represent the number of ways to choose a subset of elements from a larger set and have important combinatorial interpretations. In the setting of higher order derivatives, they serve to summarize the product of sequential integers in a concise form, linking recursive multiplicative structures to combinatorial expressions.

Mathematical Induction

Mathematical induction is a proof technique often used to establish the validity of statements for all natural numbers. In the context of this problem, induction is used to rigorously prove that the formula for the k-th derivative holds for every positive integer k by verifying the base case and then showing that if the formula holds for the k-th derivative, it also holds for the (k+1)-th derivative.

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