A proof of Leibniz’s Rule Leibniz’s Rule says that if ƒ is
Chapter , Problem 3AAE(choose chapter or problem)
A proof of Leibniz's Rule Leibniz's Rule says that if f is continuous on [a, b] and if \(u(x)\) and \(v(x)\) are differentiable functions of x whose values lie in [a, b], then
\(\frac{d}{d x} \int_{u}^{v} f(t) d t=f(v(x)) \frac{d v}{d x}-f(u(x)) \frac{d u}{d x}\)
Prove the rule by setting
\(g(u, v)=\int_{u}^{v} f(t) d t, u=u(x), v=v(x)\)
and calculating \(d g / d x\) with the Chain Rule.
Equation Transcription:
Text Transcription:
u(x)
v(x)
d/dx integral _u ^v f(t) dt = f(v(x)) dv/dx - f(u(x)) du/dx
g(u, v)= integral _u ^v f(t) dt, u=u(x), v=v(x)
dg/dx
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