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First principles and the Quotient Rule are integral parts of calculus-the branch of mathematics dealing with the rates of change of quantities, as well as the length, area, and volume of objects.
u(x)
(assuming v(x)=0)
,then
if f(x)=
v(x)
Since u(x) and v(x) which make up the function are different,
u'(x)
u(x)
[
f '(x)=
]
differentiate functions expressed in the form:
[
f '(x)=
]'
v(x)
v'(x)
f(x)=
u(x)
u(x+h)
-
v(x)
by first principles, u(x+h)-u(x)= change in the value u, or u'(x)
v(x)
v(x+h)
f '(x)=
thus,
(1+h)-1
In other words, it is used to differentiate fractions.
v(x)u(x+h)-u(x)v(x+h)
here we add[-u(x)v(x)+u(x)v(x)
which is 0. However, it facilitates factorisation in the next few steps,
=
v(x+h)v(x)
h
v(x)u(x+h)-u(x)v(x)+u(x)v(x)-u(x)v(x+h)
Splitting the fraction allows us to then factorise
v(x)u'(x)-u(x)v'(x)
=
v(x+h)v(x)
h
v(x)v(x)
or simply
v(x)u(x+h)-u(x)v(x)
u(x)v(x+h)+u(x)v(x)
-
vu'-uv'
h
=
2
v(x+h)v(x)
v
u(x+h)-u(x)
[u(x+h)-u(x)]
now, using the identity u'(x)=
v(x+h)-v(x)
v(x)
-
u(x)
h
=
(also true for v(x))
h
v(x+h)v(x)
thus, as h approaches 0
v(x)u'(x)-u(x)v'(x)
=
as h (change in value of x) is incremental, we can find the equation of the instantaneous change in value of u(x) and v(x) by using limits to 0
v(x+h)v(x)