Deriving the Power Rule using Binomial Theorem

This derivation goes to show once again the pervasiveness of mathematical concepts in different areas of the field. While Euler’s Formula does a better job of doing that, this simple derivation will suffice.

The Power Rule is often the first Derivative Rule many Calculus students will learn. It is commonly used to calculate the derivatives of polynomial functions, as well as reciprocal functions and nth roots of functions.

There are numerous and varied derivations and proofs on the internet that show the derivation of the Power Rule, but the one I will focus on involves using the Binomial Theorem through the First Principle or Difference Quotient.

Power Rule Binomial Theorem 3

When trying to come up with this derivation, I tried to avoid expanding the Summation. But evaluating the limit alongside manipulating the summation made it harder to see a clear set of steps to derive the Power Rule.

This proof makes you wonder (at least me) how Isaac Newton (or Gottfried Liebniz, but nobody really cares about him…) derived this Derivative Rule. Was it through observing the patterns in different polynomials? Was it using this exact derivation? Or was it…


Interestingly, the Power Rule was actually created by Liebniz rather than Newton, as commonly believed. Newton’s method of calculating polynomial derivatives was more complicated, thus being the reason why we continue using Liebniz’ notation. Newton and Liebniz were both contemporaries in the 17th century, with Newton being English and Liebniz being German. Newton had made his discoveries in 1666 and had them published in 1693, while Liebniz had made his in 1676 and published in 1684, sparking much controversy. This controversy was taken to court as it had become a matter of national pride for both countries. Being credited as the country for first inventing Calculus – then considered the forefront of mathematical innovation – would be immensely advantageous. As a result, it was taken to court under the mediation of the Royal Society (a society for the sciences – still present today). Coincidentally, Isaac Newton was the society’s president, thus leaving poor Gottfried Liebniz with charges of plagiarism. This story reveals the false perceptions we have of those in history.


Who would have considered Newton as someone who abused their own authority? It is similar to wondering if Leonardo Da Vinci was a bigot or Ludwig van Beethoven a drunkard.

If it is any consolation, England was far behind Europe for most of the 18th century in terms of mathematical innovation due to Newton’s complex notation.

In conclusion (a terrible way to end, but it gets the job done nonetheless), this specific derivation of the Power Rule reveals the interconnectedness of mathematical fields. The Binomial Theorem is useful in numerous situations and contexts, and this derivation proves so.

Works Cited

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Proving Binomial Theorem using Mathematical Induction

The Binomial Theorem is the perfect example to show how different streams in mathematics are connected to one another: its coefficients have combinatorial roots and can be traced to terms in Pascal’s Triangle, and expansion of binomials to different orders of power can describe probability and combination distributions. The combinatorial proof as under requires no need for proving again, but after learning a method called Mathematical Induction from incessant internet browsing on a late Saturday night, I thought, why not give it a shot?

Mathematical Induction is a method of mathematical proof used to prove an expression true for all natural numbers. The steps are as under:

  1. State the proposition P(n) that needs proving.
  2. The Basis: Show P(n) is true, when n=1.
  3. The Inductive Step:
    1. Assume n=k
    2. If P(k) is true, show that P(k+1) is true
  4. If P(k+1) is true, therefore P(n) is true.

Binomial Theorem Proof w. steps 3 - Copy

(Side-note: It’s not everyday you get to use Q.E.D.)