Songs and Lyrics (Lehrer)/The Derivative Song (dy/dx)

The Derivative Song (dy/dx)

words by Tom Lehrer
music:	"There'll Be Some Changes Made" by W. Benton Overstreet (1921) (public domain)

You take a function of x and you call it y
Take any x-nought that you care to try
You make a little change and call it delta x
The corresponding change in y is what you find nex'
And then you take the quotient and now carefully
Send delta x to zero, and I think you'll see
That what the limit gives us, if our work all checks,
Is what we call dy/dx,
It's just dy/dx.

The Derivative Song

words by Tom Lehrer
music:	"There'll Be Some Changes Made" (public domain) by W. Benton Overstreet (original lyrics by Billy Higgins)

	caption on screen
You take a function of x and you call it y	$y=f(x)$
Take any x-nought that you care to try	$y_{0}=f(x_{o})$
You make a little change and call it delta-x	$\Delta {x}=x-x_{0}$
The corresponding change in y is what you find nex'	$\Delta {y}=y-{y_{0}}$
And then you take the quotient and now carefully	${\frac {\Delta {y}}{\Delta {x}}}={\frac {y-y_{0}}{x-x_{0}}}$
Send delta-x to zero and I think you'll see	$\Delta {x}\to 0$
That what the limit gives us if our work all checks	$\lim _{\Delta {x}\to 0}{\frac {\Delta {y}}{\Delta {x}}}$
Is what we call dy/dx	$\lim _{\Delta {x}\to 0}{\frac {\Delta {y}}{\Delta {x}}}={\frac {dy}{dx}}$
It's just dy/dx	${\frac {dy}{dx}}$

The Derivative Song (dy/dx)

words by Tom Lehrer
music:	"There'll Be Some Changes Made" by W. Benton Overstreet (1921) (original lyrics by Billy Higgins)

You take a function of x and you call it y
Take any x-nought that you care to try
You make a little change and call it delta x
The corresponding change in y is what you find nex'
And then you take the quotient and now carefully
Send delta x to zero, and I think you'll see
That what the limit gives us, if our work all checks,
Is what we call dy/dx,
It's just dy/dx.

Note: should be played as

$pf = \markup {\center-align \concat { \dynamic p - \dynamic f } } music = { \language "english" { \new Staff << \set Staff.midiInstrument = "violin" { \new Voice << \relative c' { \autoBeamOff \set Score.tempoHideNote = ##t \tempo 4 = 120 \clef treble \key bf \major \time 2/2 \partial 16*5 \stemUp d16 cs8. d16 | \repeat volta 2 { a'8. a16 a8 a ~ a g4 d8 | a'8. a16 ~ a8 a ~ a2 | } } \addlyrics { You take a func -- tion of x and you call it y. } >> } { \new PianoStaff << { \new Staff << \set Staff.midiInstrument = "piano" { \new Voice << \relative c' { \clef treble \key bf \major \time 2/2 \partial 16*5 \stemUp <d bf>16[ \pf <cs bf>8. <d bf>16] \repeat volta 2 { a'8.[ a16 a8 a] ~ a8.[ g16 cs,8. d16] | a'8.[ a16 a8 a8] ~ a2 | } } >> } { \new Voice << \relative c' { \clef treble \key bf \major \time 2/2 \partial 16*5 s16 s4 | \repeat volta 2 { \stemDown b4 b8[ b] ~ b4 b | <f' b,> <f b,>8[ <f b,>] ~ <f b,>2 | } } >> } >> } { \new Staff << \set Staff.midiInstrument = "piano" { \new Voice << \relative c { \clef bass \key bf \major \time 2/2 \partial 16*5 r16 r4 | \repeat volta 2 { \stemUp <d g,> -> <d g,> <d g,> <d g,> | <d g,> <d g,> r8. g16([ fs8. f16]) | } } >> } { \new Voice << \relative c { \clef bass \key bf \major \time 2/2 \partial 16*5 s16 s4 | \repeat volta 2 { s1 | s2 \stemDown g2 | } } >> } >> } >> } >> } } \header { tagline = "" % no footer piece = CHORUS } \score { \music \layout { \context { \Staff \consists Measure_spanner_engraver } } } \score { \unfoldRepeats { \music } \midi { } }$