Leibniz Rule
Leibniz rule generalizes the product rule of differentiation. The leibniz rule states that 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. The leibniz rule is \((f(x).g(x))^n = \sum^nC_rf^{(n - r)}(x).g^r(x)\).
The leibniz rule can be applied to the product of multiple functions and for numerous derivatives. Let us understand the different formulas and proof of leibniz rule.
| 1. | What Is Leibniz Rule? |
| 2. | Proof of Leibniz Rule |
| 3. | Examples on Leibniz Rule |
| 4. | Practice Questions on Leibniz Rule |
| 5. | FAQs on Leibniz Rule |
What Is Leibniz Rule?
The Leibniz rule generalizes the product rule of differentiation. The leibniz rule states that 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. These functions can be polynomial functions, trigonometric functions,exponential functions, or logarithmic functions. Let us generalize the leibniz rule with the below formula.
\((f(x).g(x))^n = \sum^nC_rf^{(n - r)}(x).g^r(x)\)
Here \(^nC_r = \frac{n!}{r!.(n - r)!}\). and n! = 1 × 2 v 3 × 4 × ....(n - 1) × n.
The leibniz rule is primarily used as the derivative of the product of two functions. The leibniz rule for the first derivative of the product of the functions f(x), and g(x) is equal to 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). .
(f(x).g(x))' = f'(x).g(x) + f(x).g'(x)
OR
\(\dfrac{d}{dx}.f(x).g(x) = g(x).\dfrac{d}{dx}.f(x) + f(x).\dfrac{d}{dx}.g(x)\)
Further we can use the leibniz rule to find the second derivative of the product of two functions. The leibniz rule for the second derivative of the product of the functions f(x) and g(x) can be understood to be similar to the binomial expansion of two terms of second degree. Let us also check the second derivative of the product of the functions.
(f(x).g(x))'' = f''(x).g(x) + 2f'(x).g'(x) + f(x).g'(x)
OR
\(\dfrac{d^2}{dx^2}.f(x).g(x) = g(x).\dfrac{d^2}{dx^2}.f(x) + 2\dfrac{d}{dx}.f(x).\dfrac{d}{dx}.g(x)+f(x).\dfrac{d^2}{dx^2}.g(x)\)
The leibniz rule can be applied to the product of multiple functions and for numerous derivatives. The leibniz rule can be proved using mathematical induction.
Proof of Leibniz Rule
The leibniz rule can be proved with the help of mathematical induction. Let f(x) and g(x) be n times differentiable functions. Applying the initial case of mathematical induction for n = 1 we have the following expression.
(f(x).g(x))' = f'(x).g(x) + f(x).g'(x)
Which is the simple product rule and it holds true for n = 1. Let us assume that this statement is true for all n > 1, and we have the below expression.
\((f(x).g(x))^n = \sum^nC_rf^{(n - r)}(x).g^r(x)\)
Further we have the following expression for n + 1.
\((f(x).g(x))^{n+1} = ^n\sum_{r = 0} C^n_rf^{(n - r)}(x).g^r(x)\)
\(=^n\sum_{r = 0} .^nC_r.f^{(n + 1 - r)}(x).g^r(x) + ^n\sum_{r = 0}.^nC_r.f^{(n - r)}(x).g^{r + 1}(x)\)
\(=^n\sum_{r = 0} .^nC_r.f^{(n + 1 - r)}(x).g^r(x) + ^{n+1}\sum_{r = 1}.^nC_{r-1}.f^{(n+1 - r)}(x).g^r(x)\)
\(=^nC_0.f^{n + 1}(x).g(x) +^n\sum_{r = 1} .^nC_r.f^{(n + 1 - r)}(x).g^r(x) + ^n\sum_{r = 1}.^nC_{r-1}.f^{(n+1 - r)}(x).g^r(x) +^nC_n.f(x).g^{n + 1}(x)\)
\(=f^{n + 1}(x).g(x) +^n\sum_{r = 1} .[^nC_r+ ^nC_r].f^{(n + 1 - r)}(x).g^r(x) +f(x).g^{n + 1}(x)\)
\(=f^{n + 1}(x).g(x) +^n\sum_{r = 1} .^{n+1}C_r.f^{(n + 1 - r)}(x).g^r(x) +f(x).g^{n + 1}(x)\)
\(=^{n+1}\sum_{r = 0} .^{n+1}C_r.f^{(n + 1 - r)}(x).g^r(x)\)
Therefore by the principal of mathematical induction the above expression holds true for n + 1. Hence it is true for every positive integral value of n.
Related Topics
The following topics help for a better understanding of the leibniz rule.
FAQs on Leibniz Rule
What Is Leibniz Rule?
The leibniz rule states that 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. The Leibniz Rule generalizes the product rule of differentiation. The generalized expression for the leibniz rule for the nth derivative of two functions is \((f(x).g(x))^n = \sum^nC_rf^{(n - r)}(x).g^r(x)\).
How Do We Use Leibniz Rule?
The generalized formula of leibniz rule \((f(x).g(x))^n = \sum^nC_rf^{(n - r)}(x).g^r(x)\) can be simplified to find the first derivative of the product of two functions as (f(x).g(x))' = f'(x).g(x) + f(x).g'(x), and the second derivative of the product of two functions as (f(x).g(x))'' = f''(x).g(x) + 2f'(x).g'(x) + f(x).g'(x). The leibniz rule can be used for the product of two or more functions, and for n derivatives of the product of the functions.
Why Do We Use Leibniz Rule?
The leibniz rule is used to find the first, second or the derivative of the product of two or more functions. The leibniz rule for the first derivative of the product of two functions is (f(x).g(x))' = f'(x).g(x) + f(x).g'(x), and the leibniz rule for the second derivative of the product of two functions is (f(x).g(x))'' = f''(x).g(x) + 2f'(x).g'(x) + f(x).g'(x).
What Is Leibniz Rule For the First Derivative of Two Functions?
The leibniz rule for the first derivative of the product of the functions f(x) and g(x) is (f(x).g(x))' = f'(x).g(x) + f(x).g'(x) or \(\dfrac{d}{dx}.f(x).g(x) = g(x).\dfrac{d}{dx}.f(x) + f(x).\dfrac{d}{dx}.g(x)\), or \(\dfrac{d}{dx}.f(x).g(x) = g(x).\dfrac{d}{dx}.f(x) + f(x).\dfrac{d}{dx}.g(x)\).
What Is Leibniz Rule For the Second Derivative of Two Functions?
The leibniz rule for the second derivative of the product of the functions f(x) and g(x) is (f(x).g(x))'' = f''(x).g(x) + 2f'(x).g'(x) + f(x).g'(x), or \(\dfrac{d^2}{dx^2}.f(x).g(x) = g(x).\dfrac{d^2}{dx^2}.f(x) + 2\dfrac{d}{dx}.f(x).\dfrac{d}{dx}.g(x)+f(x).\dfrac{d^2}{dx^2}.g(x)\).