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The Quotient Rule (By First Principles)

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.

Lyndon Teng 12P

u(x)

(assuming v(x)=0)

Start Here

,then

if f(x)=

v(x)

The quotient rule is used to:

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

The Quotient Rule:

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)

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