   Chapter 2.3, Problem 105E

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# An easy proof of the Quotient Rule can be given if we make the prior assumption that F ′ ( x ) exists, where F = f / g . Write f = F g ; then differentiate using the Product Rule and solve the resulting equation for F ′ .

To determine

To differentiate: f=Fgusing the product rule and solve the resulting equation for F’ by the prior assumption that F'x exist, where F=fg.

Explanation

1) Concept:

Product rule: If u and v are both differentiable, then

ddxuxvx=uxddxvx+vxddxux

2) Given:

f=Fg

3) Calculations:

Assumption: F'x exist, where F=fg then f=Fg

Differentiate f=Fg using the product rule, since F'x we can use product rule with no problems.

ddxf= ddx Fg= Fddxg+g ddxF

This can be written as,

f'=  F g'+g F'

Subtract F g' from both sides,

g

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