3.1 Functions, 3.1 Relations and Functions, Difference Quotient

Difference Quotient: Rational Function

Example:  Find the difference quotient for f(x)={4x}/{x+5}

The Difference Quotient:{f(x+h)-f(x)}/h

Solution:

The Difference Quotient Formula
{f(x+h)-f(x)}/h
Write the difference quotient for the given function
={{4(x+h)}/{(x+h)+5}-{4x}/{x+5}}/h
Use the distributive property
={{4x+4h}/{x+h+5}-{4x}/{x+5}}/h
Simplify the complex fraction by multiplying the numerator and denominator by the common denominator
={{4x+4h}/{x+h+5}-{4x}/{x+5}}/h {{(x+5)(x+h+5)}/1}/ {{(x+5)(x+h+5)}/1}
Distribute the common denominator to each fraction in the numerator.
={{(4x+4h)(x+5)(x+h+5)}/{x+h+5}-{4x(x+5)(x+h+5)}/{x+5}} /{h(x+5)(x+h+5)}
Cancel the common factor
={{(4x+4h)(x+5)}-{4x(x+h+5)}} /{h(x+5)(x+h+5)}
 Multiply the expression in the numerator
={4x^2+20x+4xh+20h-4x^2-4xh-20x} /{h(x+5)(x+h+5)}
 Combine like terms
={20h} /{h(x+5)(x+h+5)}
 Cancel a common h from the numerator and denominator
={20} /{(x+5)(x+h+5)}