welcome to this video friends

and today i m going to talk about

an important topic form engineering mathematics

"Leibnitz Theorem"

friends first of all we'll talk about the statement of "Leibnitz Theorem"

so the statement of "Leibnitz Theorem" says

if we have two functions

first function is this and second function is this

ok

and suppose both of these are the functions of 'X"

and both functions are such that

that

we can find find the "nth" derivatives of both functions than

how we'll find the "nth" derivative of product of "u" and "v"

This is all "Leibnitz" about.

when we will find the nth derivative of this

then according to leibnitz theorm it says

we will write it with the help of leibnitz theorem as

now let's prove it

now we will prove this with mathmatical

induction

suppose the product of u and v is y

we have simply applied the chain rule of differentiation here

first we will talk about this part

we get

and this will be

now you see that this part

both are similar

so we add them

And we can write it as if we want

so friends you see that leibnitz is true for

now will solve it further solve it and see

so friends we will differentiate this expression again and see

we get

we get y(m+1)

when we differentiate this

when we differentiate this part we get

we are applying chain rule at this

on differentiating this part we get

we keep it as it is

when we differentiate this it becomes

we left v1 here

in the next step we will differentiate v1 and keep these terms as it is

now we will talk about this term

and we will apply chain rule here

mc2 is a constant part so it will remain as it is

in the next part we shall leave Um+2

and will differentiate V2

and it will further go like this

so friends you are noticing here

that there are some pairs forming

similarly from next two terms we get

so taking parts which are common into these terms

and these are our common terms so

we get here

similarly we get here

is

so we can write these expressions as