Leibnitz Theorem: Definition, Formula, Derivation, & Solved Questions

Leibniz Theorem, sometimes known as the Leibniz Rule, is a generalisation of the product rule of differentiation. If two functions f(x) and g(x) are differentiable n times individually, then their product f(x).g(x) is also differentiable n times, according to the Leibniz theorem. The procedure of determining the derivative of an antiderivative is described in this theorem. The term "successive differentiation" is also used to describe it. The Leibniz rule can be used to calculate the product of many functions as well as a variety of derivatives. Let us look at the various formulations and proofs for the Leibniz rule.

Key Takeaways: Leibnitz Theorem, Derivative, Differentiation, Antiderivative, Function, Integration, Successive Differentiation, Binomial Expansion

---

Leibnitz Theorem Definition

[Click Here for Sample Questions]

The Leibnitz rule is a generalisation of the product differentiation rule. If two functions f(x) and g(x) are differentiable n times individually, then their product f(x)g(x) is also differentiable n times, according to the Leibniz rule. Polynomial functions, trigonometric functions, exponential functions, and logarithmic functions are examples of these functions.

The derivative of the product of two functions is where the Leibniz rule comes in handy. The sum of the product of the first derivative of the function f(x) and the function g(x) and the product of the first derivative of the function g(x) and the function f(x) is equal to the first derivative of the product of the functions f(x) and g(x) (x). We can generalize the Leibniz rule with the below-mentioned formula.

Leibnitz Theorem

Read More: Derivatives of Function in Parametric Form

---

Leibnitz Theorem: Formula

[Click Here for Sample Questions]

Assume you have two functions, u(t) and v(t), with derivatives up to the nth order. Let's have a look at the derivative of the product of these two functions now.

The first derivative could be expressed as follows:

(uv)’ = u’v+uv’

The second derivative is obtained by differentiating the above expression once again.

(uv)’’

= [(uv)’]’

= (u’v+uv’)’

= (u’v)’+(uv’)’

= u′′v + u′v′ + u′ v′ + uv′′

= u′′v + 2u′v′ + uv′′

We can also find the third derivative in the same way.

(uv)′′′

= [(uv)′′]′

= (u′′v + 2u′v′ + uv′′)′

= (u′′v)′ + (2u′v′)′ + (uv′′)′

= u′′′v + u′′v′ + 2u′′v′ + 2u′v′′ + u′v′′ + uv′′′

= u′′′v + 3u′′v′ + 3u′v′′ + uv′′′

When these formulations are compared, they appear to be quite similar to binomial expansion raised to the exponent. We may obtain the formula for the nth order of the derivative product of two functions by considering the components with zero exponents, u₀ and v₀, which correspond to the functions u and v themselves.

The Leibnitz theorem can be used to calculate the product of many functions as well as a variety of derivatives. Mathematical induction can be used to prove the Leibnitz theorem.

Read More: Inverse Process of Differentiation

---

---

Proof of Leibnitz Theorem

[Click Here for Sample Questions]

Assume that the functions u(t) and v(t) have (n+1)th order derivatives. The derivative of (n+1)th order can be expressed in the following way using the recurrence relation:

When we differentiate, we get:

Because the limits for both sums are the same, the summation on the right side can be combined to make a single sum. Let us now consider an intermediate index, such as 1≤ m≤ n. When i = m, the first term can be expressed as follows:

When i = m-1, the second term will be:

When these two terms are added together, we get:

The concept of combinatorics tells us that:

We can write the total of these two terms when i = m and when i = m-1, as follows based on the above concept:

When the value of m goes from 1 to n, we can see that this created combination will cover all the terms from i = 1 to i = n, but not i = 0 in the first term and i = 1 in the second term, which are equivalent.

As a result, the resultant derivative of the (n+1)th order product of two functions is:

Hence Leibnitz theorem gets proved.

Read More: Differential Equation

---

Applications of Leibnitz Theorem

[Click Here for Sample Questions]

• The combination of the Leibniz Rule for R3 with Reynold's first Transport Theorem yields the second Reynold's Transport Theorem.
• The Leibniz formula computes the derivative of the product of two functions of the nth order and serves as a link between integration and differentiation.

Read More: Definite, Indefinite and Methods of Integration

---

Things to Remember

• The Leibnitz theorem asserts that if two functions, a(x) and b(x), are independently differentiable, then their combination a(x)b(x) is also n times differentiable.
• The Leibnitz theorem, commonly known as sequential differentiation, describes the process of determining an antiderivative's derivative.
• The product differentiation rule is a generalisation of the Leibnitz rule. These functions include polynomial functions, trigonometric functions, exponential functions, and logarithmic functions.
• The derivative on the nth order of the product of two functions can be stated with the help of a formula- according to the theorem.
• The Leibnitz theorem can be used to calculate the product of many functions as well as a variety of derivatives.

Also Read:

---