Page:Tom Lehrer song lyrics (website).pdf/16

THE DERIVATIVE SONG

words by Tom Lehrer

music: "There'll be Some Changes Made" (public domain)
music: "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	${\frac {dy}{dx}}$