Chapter 4
Differential Calculus and Its Uses





4.1 Derivatives and Graphs

4.1.4 The Derivative of the Reciprocal Function

If we set u = 1 s , then the difference quotient has the form

Δ ⁢ u Δ ⁢ s = 1 s + Δ ⁢ s - 1 s Δ ⁢ s
  = 1 Δ ⁢ s ( 1 s + Δ ⁢ s - 1 s ) .

If we rewrite 1 s + Δ ⁢ s - 1 s with a common denominator, we obtain

Δ ⁢ u Δ ⁢ s = 1 Δ ⁢ s s - ( s + Δ ⁢ s ) s ( s + Δ ⁢ s )
  = 1 Δ ⁢ s - Δ ⁢ s s ( s + Δ ⁢ s )
  = - 1 s ( s + Δ ⁢ s ) .

As Δ ⁢ s approaches zero, the difference quotient approaches - 1 s 2 . Thus, the derivative of the reciprocal function is the negative of the reciprocal square function:

d d ⁢ s 1 s = - 1 s 2 .
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