Cheatsheet: The power rule from geometry

The power rule

d/dt( t^n ) = n · t^(n-1)

The n counts the faces that grow when you nudge one dimension; the t^(n-1) is the size of each face.

Where it comes from (geometry)

Function	Picture	Growth when side -> t + dt	Derivative
t^2	square, side t	2 strips (t·dt each) + corner (dt^2)	2t
t^3	cube, side t	3 slabs (t^2·dt each) + smaller terms	3t^2

The corner / edge terms (dt^2, dt^3) vanish faster than the strips as dt -> 0, so they drop; the strips survive.

Works for any power

1/t  = t^(-1)   ->  d/dt = -1·t^(-2) = -1/t^2
√t   = t^(1/2)  ->  d/dt = (1/2)·t^(-1/2) = 1/(2√t)

Negative and fractional exponents follow the same rule.

Two linearity rules

Rule	Statement	Why
Constant multiple	d/dt(c·f) = c·d/dt(f)	Stretching a graph vertically by c scales every slope by c.
Sum	d/dt(f+g) = d/dt(f)+d/dt(g)	Rates of a sum are the sum of the rates.

A constant has derivative 0 (it never changes).

Worked polynomial

d/dt( 3t^4 + 2t^2 - 7 )
  = 3·4t^3  +  2·2t  +  0
  = 12t^3   +  4t

Three mechanical lines, no binomial expansion.

Why it matters for AI

These rules are what automatic differentiation applies to compute gradients, chaining the derivative of each elementary operation through the network. The power rule is everywhere because squaring is: mean squared error has terms (prediction - target)^2, whose derivative is 2·(prediction - target), so the gradient is proportional to the error.

Pitfalls to dodge

• Memorizing without the picture. n = faces that grow, t^(n-1) = each face’s size; rederive from the square/cube.
• Keeping the corner term. The dt^2/dt^3 pieces vanish; the strips/slabs survive.
• Forgetting a constant’s derivative is 0. Shifting a graph up/down changes no slopes.
• Thinking it is whole-powers only. Holds for negative and fractional n too.

The one-line version

The power rule d/dt(t^n) = n·t^(n-1) is just “how many faces grow, times each face’s size” when you nudge a cube’s side, and the constant-multiple and sum rules let it differentiate any polynomial on sight.