[Verse 1]
When functions dance together, add or multiply
There's secrets in their movement, rules that never lie
Sum rule keeps it simple, just add up each part
Prime of f plus g equals f-prime plus g-prime, that's the start
[Chorus]
Differentiate with precision, follow every law
Sum and constant, product, quotient, chain rule conquers all
f-prime-g plus f-g-prime, that's the product way
Chain rule multiplies the rates through every single relay
[Verse 2]
Constants are companions, they never change their face
Multiply by c just pulls it to the front without a trace
But products need attention, both functions get their turn
First times second's derivative, plus second times first's return
[Chorus]
Differentiate with precision, follow every law
Sum and constant, product, quotient, chain rule conquers all
f-prime-g plus f-g-prime, that's the product way
Chain rule multiplies the rates through every single relay
[Bridge]
Quotient rule's a fraction, numerator's got the trick
f-prime-g minus f-g-prime, divided by g-squared quick
But chain rule is the master, composition's secret key
Outer prime of inner times the inner's velocity
[Verse 3]
Like runners in a relay race, each function passes on
Their rate of change keeps flowing until the race is won
One slow link breaks the tempo, makes the whole thing crawl
Chain rule captures everything, the greatest rule of all
[Final Chorus]
Differentiate with precision, follow every law
Sum and constant, product, quotient, chain rule conquers all
Backpropagation's backbone, substitution's might
Chain rule is the foundation that makes calculus take flight
[Outro]
Five rules to rule them all, derivatives made clear
Mathematics flows like music when the patterns appear