A change passes through two machines. Each one stretches it, so the stretch factors multiply.
WHAT HAPPENS IF…Can the composition have derivative zero even when the inner derivative is nonzero?
Interactive exploration
Preparing your experiment…
Connect the picture to the ideadxdsin(kx2)=cos(kx2)⋅2kx
Let u = kx² and y = sin u. A small change in x first changes u at rate 2kx; then u changes y at rate cos u. Multiplying these rates gives the derivative of the composition.
A detail that matters
The chain rule multiplies derivatives; it does not add them.