9.6: The Difference Quotient
- Page ID
- 174339
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Given a function \(f\), the difference quotient of \(f\) is the expression\[\dfrac{f(x+h)-f(x)}{h}. \nonumber \]
With difference quotients, you will commonly need to multiply the numerator and denominator of an expression by either
- the LCD of all fractions within a compound fraction (a.k.a. simplifying compound fractions), or
- the conjugate of either the numerator or the denominator (a.k.a. rationalizing).
When doing so, knowing what to distribute and what not to distribute is imperative.
Simplifying Compound Fractions
The entire point of multiplying the numerator and denominator of your compound fraction by the LCD of all the minor fractions is to "get rid of" denominators in those minor fractions; however, with difference quotients, it is common that the denominator of the entire (major) fraction is only \( h \). In this case, do not distribute out the denominator! Doing so will complicate the mathematics, and you will lose visibility of factors that cancel.
Rationalizing
In a difference quotient, the entire point of multiplying the numerator and denominator by the conjugate of either the numerator or the denominator is to clear radicals. This will only happen with the conjugate pairs. So, definitely distribute the conjugate pairs, but do not distribute the non-conjugates.