 # Leibniz Rule

Leibniz Rule is the rule defined for derivative of the antiderivative. As per the Leibniz rule, the derivative on the $$n^{th}$$ order of the product of two functions can be expressed with the help of a formula.

German philosopher and mathematician Gottfried Wilhelm Leibniz used the symbols $$dx$$ and $$dy$$ to represent infinitely small (or infinitesimal) increments of x and y, respectively, just as $$\Delta x$$ and $$\Delta y$$ represent finite increments of $$x$$ and $$y$$, respectively.

Gottfried Wilhelm von Leibniz invented the calculating machine in 1671, which was a significant advance in mechanical calculating. The rules for calculus were ﬁrst laid out in Gottfried Wilhelm Leibniz’s 1684 paper.

Index

## Leibniz Theorem

Leibniz rule basically generalizes the product rule. It states that u and v are -times differentiable functions, then the product uv is also n-times differentiable and its nth derivative is given by

$${(uv)’}^{n} = {\sum^{n}_{i=0}} ^nC_i {u’}^{(n-i)} {v’}^i$$

On substituting n=1 in this formula we get product rule
$$(uv)’ = u’v + uv’$$

Note: Another similar rule in calculus is quotient rule.

### Derivation

Let us consider two functions $$u(x)$$ & $$v(x)$$, and they have the derivatives upto the $$n^{th}$$ order.

Now, the first order of derivative can be written as

$$(uv)’ = u’v + uv’$$

Now, the second order of derivative will be

$$(uv)” = ((uv)’)’ = (u’v + uv’)’$$

= $$u”v + u’v’ + u’v’ + uv”$$

= $$u”v + 2(u’v’) + uv”$$

And the third derivative will be,

$$(uv)’’’ = (uv)^{’^3}$$

= $$(u”v + 2(u’v’) + uv”)’$$

= $$u’’’v + u’’v’ + 2u’’v’ + 2u’v’’ + u’v’’ + uv’’’$$

= $$u’’’v + 3u’’v’ + 3u’v’’ + uv’’’$$

Now, taking the above in consideration we can find an formula for $$n$$ terms.

$$(uv)’^{n} = \sum_{i = 0}^{n} u^{(n-i)}v^i$$

= $$(u”v + 2(u’v’) + uv”)’$$

= $$u’’’v + u’’v’ + 2u’’v’ + 2u’v’’ + u’v’’ + uv’’’$$

= $$u’’’v + 3u’’v’ + 3u’v’’ + uv’’’$$

Now, taking the above into consideration, we can find a formula for $$n$$ terms.

$${(uv)’}^{n} = {\sum^{n}_{i=0}} ^nC_i {u’}^{(n-i)} {v’}^i$$

This formula is known as the Leibniz Rule.

### Proof

This formula can be proved by Principle of Mathematical Induction.

Let us consider two functions $$u(x)$$ & $$v(x)$$, and they have the derivatives upto the $$(n+1)^{th}$$ order.

$${y’}^{(n+1)} = ({y’}^n)’$$

= $$[ {\sum^{n}_{i=0}} ^nC_i {u’}^{n – i} {v’}^{i}]’$$

= $$[ {\sum^{n}_{i=0}} ^nC_i {u’}^{n – i + 1} {v’}^{i}] + … (i)$$

$$[ {\sum^{n}_{i=0}} ^nC_i {u’}^{n – i} {v’}^{i + 1}] … (ii)$$

Now, let us combine the summation on right side in a single sum, as both holds the same limits.

Let $$\exists m$$ such that $$1 \leq m \leq n$$.

So, when $$i = m$$, (i) becomes

$$^nC_m {u’}^{n – m + 1} {v’}^{m}$$ … (iii)

And, the second term $$i + 1 = m$$ (or) $$i = m – 1$$, becomes

$$^nC_{m-1} {u’}{n – (m – 1)} {v’}^{(m- 1) + 1}$$

$$\Rightarrow ^nC_{m-1} {u’}^{n – m + 1} {v’}^{m} … (iv)$$

Adding (iii) & (iv) we get

$$^nC_m {u’}^{n – m + 1} {v’}^{m} + ^nC_{m-1} {u’}{n – m + 1} {v’}^{m} = [^nC_m +^nC_{m-1}] {u’}^{n – m + 1} {v’}^{m}$$

From combinatorics, we use addition and get;

$$[^nC_m +^nC_{m-1}] {u’}^{n – m + 1} {v’}^{m} = [^{n + 1}C_{m}] {u’}^{n – m + 1} {v’}^{m} … (v)$$

As, we can see that the values of $$m$$ will change from $$[1, n]$$ for $$i$$, but it will not cover the values for $$i = 0$$ in (i) & $$i = 1$$ in (ii) cases.

Hence,

When $$i = 0$$ in (i) $$\Rightarrow ^nC_0 {u’}^{n – 0 + 1} {v’}^{0} = {u’}^{n + 1} {v’}^{0} … (vi)$$

& $$i = n { \mbox{ since } ^nC_1 = ^nC_n}$$

in (ii) $$\Rightarrow ^nC_i {u’}^{n – i} {v’}^{i + 1}] = {u’}^{0} {v’}^{n+1} … (vii)$$

Hence, from the $$(n + 1)^{th}$$ derivative we got the result by adding (v), (vi) & (vii)

$$y^{n+1} = {u’}^{n + 1} {v’}^{0} + {\sum^{n}_{i=0}} [^{n + 1}C_{m}] {u’}^{n – m + 1} {v’}^{m} + {u’}^{0} {v’}^{n+1}$$

$$\Rightarrow {\sum^{n}_{i=0}} ^{n+1}C_m {u’}^{n + 1 – m} {v’}^{m}$$

Hence Proved, for $$(n + 1)$$ terms. Hence the theorem gets proved.

## Applications of Leibniz Rule

The second Reynold’s Transport Theorem is deduced from the application of the Leibniz Rule for $$\mathbb{R}^3$$ with Reynold’s first Transport theorem.

The Leibniz formula gives the derivative on $$n^{th}$$ order of the product of two functions and works as a connection between integration and differentiation.

## Examples on Leibniz Rule

Question 1. If $$f(x) = sin x$$ and $$g(x) = cos x$$
Find the coefficient of $$sin^{11} (2x)$$ of function $$f(x)g(x)$$.

Solution. Let $$h(x )= y = f(x)g(x)$$

$$y’^{22} = {\sum^{n}_{i=0}} ^{22}C_i {f(x)}’^{(22 – i)} {g(x)’}^{i}$$

$$y’^{22} = ^{22}C_0 {f(x)’}^{(22 – 0)} {g{x}’}^0 + ^{22}C_1 {f(x)’}^{(22 – 1)} {g(x)’}^{1} ….^{22}C_{22} {f(x)’}^{(22 – 22)} {g(x)’}^{22}$$

$$\Rightarrow ^{22}C_0 {sin’}^{(22 – 0)}(x) {cos’}^{0}(x) + ^{22}C_1 {sin’}^{(22 – 1)} (x) {cos’}^{1}(x) ….^{22}C_{22} {sin’}^{(22 – 22)}(x) {cos’}^{22}(x)$$

Now for $$sin^{11} (2x)$$ will be the $$12^{th}$$ term of the sequence

Therefore, $$^{22}C_{11} sin^{22 – 11}(x) cos^{11}(x)$$

$$\Rightarrow ^{22}C_{11} \frac{1}{2} sin^{11}(2x)$$

Therefore the coefficient of $$sin^{11}(2x)$$ is $$^{22}C_{11} \frac{1}{2}$$.

Question 2. Let $$f(x) = \sqrt{1 – x^2}$$ and let $${y’}^{n}$$ denote the $$n^{th}$$ derivative of $$f(x)$$ at $$x = 0$$, then find the value of $$6{y’}^{1} {y’}^{2} + 2{y’}^{3}$$

Solution. Let $$y = f(x) = \sqrt{1 – x^2}$$

We can rewrite the function as

$$y^2 = 1 – x^2$$

Differentiating both sides upto third derivative using Leibniz rule

$$(y^2)’^{3} = ^3C_0 {y’}^{3}y + ^3C_1 {y’}^{2} {y’}^{2} + ^3C_2 {y’}^{1} {y’}^{2} + ^3C_3 {y’}^{0} {y’}^{3}$$

$${(y^2)’}^{3} = 2{y’}^{3} + 6{y’}^{2} {y’}^{1} … (i)$$

& $${(1 – x^2)’}^{3} = 0 … (ii)$$

From (i) & (ii) we get,

$$2{y’}^{3} + 6{y’}^{2} {y’}^{1} = 0$$

## FAQs

What is Leibniz Theorem?

The Leibniz formula expresses the $$n^{th}$$ order of derivative of a product of two functions.

What is Leibniz’s notation?

Leibniz uses the symbols $$dx$$ and $$dy$$ to represent infinitesimal increments of $$x$$ and $$y$$, respectively, and as $$\Delta x$$ and $$\Delta y$$ represent finite increments of $$x$$ and $$y$$, respectively.

Who came up with the product rule (leibniz rule)?

The rule was given by Gottfried Wilhelm Leibniz

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