To prove quotient rule formula using the definition of derivative or limits, let the function f(x) = u(x)/v(x). = f'(x) = lim [f(x + h) - f(x)]/h h-O u(x+h)ux) vix+h) v(x) = lim h-0 u(x+h)v(x)-u(x)v(x+h) lim h-v(x)-v(x+h)
To prove quotient rule formula using the definition of derivative or limits, let the function f(x) = u(x)/v(x). = f'(x) = lim [f(x + h) - f(x)]/h h-O u(x+h)ux) vix+h) v(x) = lim h-0 u(x+h)v(x)-u(x)v(x+h) lim h-v(x)-v(x+h)
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter3: The Derivative
Section3.5: Graphical Differentiation
Problem 2E
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why do we need to add and subtract at this part in the second picture.
![To prove quotient rule formula using the definition of derivative or limits,
let the function f(x) = u(x)/v(x).
= f'(x) = lim [f(x + h) - f(x)]/h
h-0
u(x+h)
u(x)
v(x+h)
v(x)
= lim
h-0
u(x+h)v(x)-u(x)v(x+h)
= lim
h-0
h-v(x)-v(x+h)
u(x+h)v(x)-u(x)v(x+h)
= (lim
h-0
) (lim v+n))
h
h-0
u(x+h)v(x)-u(x)v(x)+u(x)v(x)-u(x)v(x+h)
= (lim
h-0
-) [1/v(x)²]
h
= [(lim
h-0
u(x+h)v(x)-u(x)v(x)Hlim
u(x)v(x+h)-u(x)v(x)
[1/v(x)?1
h
u(x+h)¬ux) )
u(x+h)-u(x)
= [v(x)(lim
h-0
-) -u(xX(lim x+h)-v(x)
-) ] [Iw(x)²]
h-0
v(x)u'(x)-u(x)v (x)
[v(x)]?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e090baf-bc58-49b6-bf20-712458fdedf0%2F494ba0a0-7f2a-416f-83c3-f57aedbbdb25%2Fy4aqjpn_processed.png&w=3840&q=75)
Transcribed Image Text:To prove quotient rule formula using the definition of derivative or limits,
let the function f(x) = u(x)/v(x).
= f'(x) = lim [f(x + h) - f(x)]/h
h-0
u(x+h)
u(x)
v(x+h)
v(x)
= lim
h-0
u(x+h)v(x)-u(x)v(x+h)
= lim
h-0
h-v(x)-v(x+h)
u(x+h)v(x)-u(x)v(x+h)
= (lim
h-0
) (lim v+n))
h
h-0
u(x+h)v(x)-u(x)v(x)+u(x)v(x)-u(x)v(x+h)
= (lim
h-0
-) [1/v(x)²]
h
= [(lim
h-0
u(x+h)v(x)-u(x)v(x)Hlim
u(x)v(x+h)-u(x)v(x)
[1/v(x)?1
h
u(x+h)¬ux) )
u(x+h)-u(x)
= [v(x)(lim
h-0
-) -u(xX(lim x+h)-v(x)
-) ] [Iw(x)²]
h-0
v(x)u'(x)-u(x)v (x)
[v(x)]?
![u(x+h)v(x)-u(x)v(x)+u(x)v(x)-u(x)v(x+h)
= (lim
h-0
:) [1/v(x)²]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e090baf-bc58-49b6-bf20-712458fdedf0%2F494ba0a0-7f2a-416f-83c3-f57aedbbdb25%2Fb3hmkvv_processed.png&w=3840&q=75)
Transcribed Image Text:u(x+h)v(x)-u(x)v(x)+u(x)v(x)-u(x)v(x+h)
= (lim
h-0
:) [1/v(x)²]
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