LLM Primer

Summary and Revision Notes

Use this page to review the chapter before doing the exercises.

Key Ideas

IdeaMeaningQuestion to ask
average slopechange in output divided by change in inputWhat happened over this interval?
derivativelocal rate of changeWhat happens near this input?
partial derivativeone-input sensitivityWhat if this input changes and others stay fixed?
gradientvector of partial derivativesWhat is the local sensitivity vector?
directional derivativechange along a chosen unit directionWhat if we move this way?
Jacobianderivative table for vector outputsHow do all outputs respond to all inputs?
Hessiansecond-derivative tableHow is the gradient changing?
chain rulelocal sensitivities multiplied through compositionHow does change pass through this path?
computation graphdependency map of values and operationsWhich values depend on which earlier values?
backpropagationchain rule applied backward through a graphHow did earlier values affect the loss?

Formulas to Remember

Average slope:

ΔyΔx\frac{\Delta y}{\Delta x}

Local approximation:

Δf≈f′(x)Δx\Delta f \approx f'(x)\Delta x

Gradient:

∇f=[∂f/∂x1,…,∂f/∂xn]\nabla f = [\partial f/\partial x_1,\ldots,\partial f/\partial x_n]

Directional derivative:

Duf(x)=∇f(x)⋅uD_u f(x) = \nabla f(x) \cdot u

Chain rule:

dydx=dydgdgdx\frac{dy}{dx} = \frac{dy}{dg}\frac{dg}{dx}

Gradient descent:

wnew=w−η∇L(w)w_{new} = w - \eta \nabla L(w)

Useful Distinctions

Do not confuseDifference
average slope and derivativeaverage slope uses two points; derivative is local at one point
derivative and partial derivativea partial derivative is for a multi-input function
gradient and descent directionthe gradient points uphill; descent uses the negative gradient
gradient and Jacobiana gradient is for scalar output; a Jacobian is for vector output
Jacobian and Hessiana Jacobian tracks first derivatives; a Hessian tracks second derivatives
chain rule and backpropagationbackpropagation is organized chain rule over a graph
local approximation and guaranteea derivative gives a nearby estimate, not a global promise

Common Traps

  • Treating a derivative as a global description of the whole function.
  • Forgetting that partial derivatives hold other variables fixed.
  • Comparing gradient components by sign instead of magnitude.
  • Moving with the gradient when the goal is to decrease loss.
  • Forgetting that a directional derivative usually uses a unit direction.
  • Thinking a zero gradient always means the best possible point.
  • Treating backpropagation as separate from the chain rule.
  • Forgetting that branch contributions are added during backpropagation.
  • Trusting an autodiff result without checking shapes and dependencies.
  • Forgetting that the Jacobian is the local linear map for vector-output changes.

Mental Model

Calculus tells how quantities change.

Training uses that information to adjust parameters.

Backpropagation is the bookkeeping that carries those local change signals through a computation graph.