[Verse 1] When z depends on x and y so tight And both are dancing through the night To find how z will change with time We need a rule that's more sublime Than simple derivatives we knew This pathway thinking pulls us through [Chorus] Every path contributes, can't ignore Dee z dee t equals so much more Partial z with partial x times dee x dee t Plus partial z with partial y times dee y dee t Chain reaction, sum the ways Through every variable that sways [Verse 2] Picture water flowing down a hill Temperature and pressure shifting still The gradient shows us every route While Jacobian makes the truth compute Each channel carries its own weight Matrix form will calculate [Chorus] Every path contributes, can't ignore Dee z dee t equals so much more Partial z with partial x times dee x dee t Plus partial z with partial y times dee y dee t Chain reaction, sum the ways Through every variable that sways [Bridge] Neural networks learn this song Backpropagation tags along Gradient flows from loss to start Chain rule plays the crucial part Del f composed with g transforms Jacobian transpose performs [Verse 3] When variables nest inside each other One small change affects another Total rate breaks down in pieces As dependency never ceases Weight each pathway by its speed Sum them up for what you need [Chorus] Every path contributes, can't ignore Dee z dee t equals so much more Partial z with partial x times dee x dee t Plus partial z with partial y times dee y dee t Chain reaction, sum the ways Through every variable that sways [Outro] Multivariable chain unfolds The story that calculus holds Every pathway tells its tale Total change will never fail
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