3.1 Functions, 3.1 Relations and Functions, Difference Quotient

Difference Quotient: Rational Function

Example:  Find the difference quotient for f(x)=1/x^2

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
={1/(x+h)^2-1/x^2}/h
Simplify the complex fraction by multiplying the numerator and denominator by the common denominator
={1/(x+h)^2-1/x^2}/h {{x^2(x+h)^2}/1}/{{x^2(x+h)^2}/1}
Distribute the common denominator to each fraction in the numerator.
={1/(x+h)^2-1/x^2}/h {{x^2(x+h)^2}/1}/{{x^2(x+h)^2}/1}
={{x^2(x+h)^2}/(x+h)^2-{x^2(x+h)^2}/x^2}/{h{x^2(x+h)^2}}
Cancel the common factor
={{x^2(x+h)^2}/(x+h)^2-{x^2(x+h)^2}/x^2}/{h{x^2(x+h)^2}}
={x^2-(x+h)^2}/{h{x^2(x+h)^2}}
Simplify by squaring the binomial
={x^2-(x+h)^2}/{h{x^2(x+h)^2}}
={x^2-(x+h)(x+h)}/{h{x^2(x+h)^2}}
={x^2-(x^2+xh+xh+h^2)}/{h{x^2(x+h)^2}}
Simplify by combining like terms and distributing the negative
={x^2-(x^2+2xh+h^2)}/{hx^2(x+h)^2}
={x^2-x^2-2xh-h^2}/{hx^2(x+h)^2}
 Combine like terms
={-2xh-h^2}/{hx^2(x+h)^2}
 Cancel a common h from the numerator and denominator
={-2x-h}/{x^2(x+h)^2}

The difference quotient for f(x)=1/x^2 is {-2x-h}/{x^2(x+h)^2}