3.1 Functions | math15fun.com

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
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	= ${-2x-h}/{x^2(x+h)^2}$