Calculus
Derivative Rules
Three rules, one picture each. Add two functions and their graphs stack, so the slopes add. Multiply them and a rectangle grows two strips and one doomed corner. Compose them and a small step is stretched twice, so the factors multiply.
u and its rise Δuv and its rise Δvu + v
The tower grows twice: once for u, once for v
the side strip v·Δuthe top strip u·Δvthe corner Δu·Δv
Each edge paints a strip as it moves; the corner is the only piece that dies
g stretches the stepf stretches thatthe whole chain
One scale for all three bars, so a longer bar really was stretched
Slope of u + v—
—
Rule
The two functions
The two factors
The chain f(g(x))
a big stepa tiny one
Values
Δu—
Δv—
Δ(u+v)—
Δ(u+v)/h—
u′ + v′—
leftover terms—
Values
u, v—
Δu, Δv—
side strip v·Δu—
top strip u·Δv—
corner Δu·Δv—
ΔA / h—
u′v + uv′—
Values
u = g(x)—
g stretches by g′(x)—
f stretches by f′(u)—
the two multiplied—
Δy / h at this step—
What to observe
- Sum. Run it and watch the tower: the top rises by Δu plus Δv, exactly, at every h. So (u+v)′ = u′ + v′ with nothing thrown away.
- Product. Shrink h: the two strips shrink like h, the corner like h². Only the strips survive, so (uv)′ = u′v + uv′, not u′v′.
- Chain. On sin(x²) at x = 1.25, f′(u) is almost zero, so the bottom bar collapses while g still stretches: a chain is as dead as itsdeadest link.
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