Leibnitz Rule Proof

Leibniz’s rule can be proven using mathematical induction. If we have two functions, f(x) and g(x), that can be smoothly changed many times (differentiable), we start by showing that the product rule holds true for n = 1:

Derivation of Leibnitz Theorem

Step 1: Base Case (n = 1)

Start with the product of two differentiable functions, f(x) and g(x), and apply the product rule:

(f(x)⋅g(x))′ = f′(x)⋅g(x) + f(x)⋅g′(x)

This is the basic product rule and serves as the base case.

Step 2: Inductive Hypothesis

Assume that Leibniz’s rule holds for some positive integer (n):

[Tex](f(x) \cdot g(x))^n = \sum_{r=0}^{n} \binom{n}{r} f^{(n-r)}(x) \cdot g^{(r)}(x) [/Tex]

This is inductive hypothesis.

Step 3: Inductive Step (n + 1)

Now, we want to show that the rule holds for n+1:

[Tex](f(x) \cdot g(x))^{n+1} [/Tex]

Using the binomial theorem, expand this expression:

[Tex](f(x) \cdot g(x))^n \cdot (f(x) \cdot g(x)) [/Tex]

Apply the inductive hypothesis to the first part and the product rule to the second part:

[Tex]\sum_{r=0}^{n} \binom{n}{r} f^{(n-r)}(x) \cdot g^{(r)}(x) \cdot (f(x) \cdot g(x))’ [/Tex]

Now, expand the product rule in the second term:

[Tex]\sum_{r=0}^{n} \binom{n}{r} f^{(n-r)}(x) \cdot g^{(r+1)}(x) + \sum_{r=0}^{n} \binom{n}{r} f^{(n-r+1)}(x) \cdot g^{(r)}(x) [/Tex]

Combine like terms:

[Tex]\sum_{r=0}^{n+1} \left(\binom{n}{r} f^{(n-r)}(x) \cdot g^{(r+1)}(x) + \binom{n}{r-1} f^{(n-r+1)}(x) \cdot g^{(r)}(x)\right) [/Tex]

Factor out terms and simplify:

[Tex]\sum_{r=0}^{n+1} \binom{n+1}{r} f^{(n+1-r)}(x) \cdot g^{(r)}(x) [/Tex]

By the principle of mathematical induction, the expression holds true for all positive integral values of \(n\). Therefore, Leibniz’s rule is proven.

Also, Check

• Euler’s Theorem
• Rolle’s Theorem and Lagrange’s Mean Value Theorem
• Derivative Rules

Leibniz’s Theorem is a fundamental concept in calculus that generalizes the product rule of differentiation and helps us find the n^th derivative of the product of two functions. It is a powerful tool in mathematical analysis, particularly when dealing with functions that change smoothly.

This theorem plays a crucial role in modeling instantaneous rates of change in various mathematical and real-world scenarios. In this article, you will learn the formula of the Leibnitz Theorem, the proof, and the derivation of the Leibnitz Theorem.