3. A proof of Leibniz's Rule Leibniz's Rule says that if f is con- tinuous on [a, b] and if u(x) and v(x) are differentiable functions of x whose values lie in [a, b], then ru(x) d dv f(t) dt = f(v(x)) dx du f(u(x)) dx dx u(х) Prove the rule by setting g(и, v) f(t) dt, и 3 и(х), v = v(x) and calculating dg/ dx with the Chain Rule.
3. A proof of Leibniz's Rule Leibniz's Rule says that if f is con- tinuous on [a, b] and if u(x) and v(x) are differentiable functions of x whose values lie in [a, b], then ru(x) d dv f(t) dt = f(v(x)) dx du f(u(x)) dx dx u(х) Prove the rule by setting g(и, v) f(t) dt, и 3 и(х), v = v(x) and calculating dg/ dx with the Chain Rule.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![3. A proof of Leibniz's Rule Leibniz's Rule says that if f is con-
tinuous on [a, b] and if u(x) and v(x) are differentiable functions
of x whose values lie in [a, b], then
ru(x)
d
dv
f(t) dt = f(v(x))
dx
du
f(u(x))
dx
dx
u(х)
Prove the rule by setting
g(и, v)
f(t) dt,
и 3 и(х),
v = v(x)
and calculating dg/ dx with the Chain Rule.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faddd048f-bb89-4267-b54c-271529ad2bd2%2Fd7fe670b-bf5c-4454-b20e-4ef7641b37ec%2Fum4q6r6.png&w=3840&q=75)
Transcribed Image Text:3. A proof of Leibniz's Rule Leibniz's Rule says that if f is con-
tinuous on [a, b] and if u(x) and v(x) are differentiable functions
of x whose values lie in [a, b], then
ru(x)
d
dv
f(t) dt = f(v(x))
dx
du
f(u(x))
dx
dx
u(х)
Prove the rule by setting
g(и, v)
f(t) dt,
и 3 и(х),
v = v(x)
and calculating dg/ dx with the Chain Rule.
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