What is the form of “Leibniz Rule” for integral of f(x,a,b) from a to b? x is variable and derivation is done in respect of a and b.Since the “Leibniz Rule” for derivation of finite...
What is the form of “Leibniz Rule” for integral of f(x,a,b) from a to b? x is variable and derivation is done in respect of a and b.
Since the “Leibniz Rule” for derivation of finite integrals is presented in literature for f(x,α) with limits as a(α) and b(α) and derivation in respect of α, I have wondered. Please help me.
1 Answer
sciencesolve | Teacher | (Level 3) Educator Emeritus
You need to remember the Leibniz rule for a differentiable function `f(v,x)` such that:
`d/(dx) int_(a(x))^(b(x)) f(v,x) dv = int_(a(x))^(b(x)) (del f(v,x))/(del x) dv`
In this case, the limits `a(x)` and `b(x)` are constants, but if `a(x)` and `b(x)` are not constants, hence, the extended Leibniz rule is the following, such that:
`d/(dx) int_(a(x))^(b(x)) f(v,x) dv = int_(a(x))^(b(x)) (del f(v,x))/(del x) dv + b'(x)(f(b(x),x)) - a'(x)(f(a(x),x))`
Hence, you may use either the extended form, or the simpler form of Leibniz rule, depending on the nature of limits `a(x)` and `b(x).`