define the Leibnitz Theorem
Answers
Answer:
Leibnitz Theorem is basically the Leibnitz rule defined for derivative of the antiderivative. As per the rule, the derivative on nth order of the product of two functions can be expressed with the help of a formula. The functions that could probably have given function as a derivative are known as antiderivatives (or primitive) of the function. The formula that gives all these antiderivatives is called the indefinite integral of the function, and such process of finding antiderivatives is called integration. Now let us discuss here the formula and proof of Leibnitz rule.
Step-by-step explanation:
Leibnitz Theorem Formula
Suppose there are two functions u(t) and v(t), which have the derivatives up to nth order. Let us consider now the derivative of the product of these two functions.
The first derivative could be written as;
(uv)’ = u’v+uv’
Now if we differentiate the above expression again, we get the second derivative;
(uv)’’
= [(uv)’]’
= (u’v+uv’)’
= (u’v)’+(uv’)’
= u′′v + u′v′ + u′ v′ + uv′′
= u′′v + 2u′v′ + uv′′
Similarly, we can find the third derivative;
(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′′′
Now if we compare these expressions, it is found very similar to binomial expansion raised to the exponent. If we consider the terms with zero exponents, u0 and v0 which correspond to the functions u and v themselves, we can generate the formula for nth order of the derivative product of two functions, in a such a way that;
Answer:
Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form where, the derivative of this integral is expressible as.
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