[Verse 1]
When you zoom into a curve until it's straight
That microscopic slope won't hesitate
To show you what the function's doing now
The tangent line reveals exactly how
[Chorus]
F prime of X equals the limit as H approaches zero
Of F of X plus H minus F of X over H, that's our hero
The difference quotient shrinking down to nothing but the truth
Instantaneous change captured, that's the derivative proof
[Verse 2]
Position turns to velocity with one swift calculation
Distance over time becomes acceleration
The ratio of infinitesimals, though Leibniz knew the score
DF over DX opens up the door
[Chorus]
F prime of X equals the limit as H approaches zero
Of F of X plus H minus F of X over H, that's our hero
The difference quotient shrinking down to nothing but the truth
Instantaneous change captured, that's the derivative proof
[Bridge]
Marginal cost from total cost
The rate of change is never lost
How fast things move at this precise moment
The slope of curves in their component
[Verse 3]
Every smooth curve hides a secret line
When intervals collapse to points so fine
The difference quotient holds the key
To rates of change for you and me
[Chorus]
F prime of X equals the limit as H approaches zero
Of F of X plus H minus F of X over H, that's our hero
The difference quotient shrinking down to nothing but the truth
Instantaneous change captured, that's the derivative proof
[Outro]
Zoom in close and see the slope
The derivative gives us hope
To understand how functions bend
The tangent line's our faithful friend