Quizmaster

Learning objectives

What you will be able to explain

Univariate rules
Compute a gradient
Gradients
Vector-valued derivatives
Gradients with matrices
Quadratic-form gradient

Section 01

Univariate rules to Gradients

Univariate rules

The chapter begins with scalar rules before generalizing dimensions.

Guided checkpoint

Match function to derivative.

Source: Sec. 5.1, pp. 141-146

Compute a gradient

The derivatives are

2x+3y

and

3x+4y

Guided checkpoint

For $f(x,y)=x^2+3xy+2y^2$ , evaluate the gradient at $(1,2)$ .

Source: Sec. 5.2, pp. 146-149

Gradients

The gradient represents the derivative as an inner product.

Guided checkpoint

For scalar $f:R^D \to R$ .

Source: Sec. 5.2, pp. 146-149

Section 02

Vector-valued derivatives to Quadratic-form gradient

Vector-valued derivatives

Jacobians linearize vector-valued maps and compose by matrix multiplication.

Guided checkpoint

For $f:R^n \to R^m$ , match object.

Source: Sec. 5.3, pp. 149-155

Gradients with matrices

Trace and differential manipulations preserve noncommuting order.

Guided checkpoint

Check the book's conventions.

Source: Secs. 5.4-5.5, pp. 155-159

Quadratic-form gradient

Both occurrences of

x

contribute; symmetry reduces this to

2Ax

Guided checkpoint

For constant $A$ , $nabla_x(x^TAx)$ equals:

Source: Sec. 5.5, pp. 158-159

Section 03

Automatic differentiation modes to Second-order Taylor model

Automatic differentiation modes

Mode choice depends on Jacobian dimensions; backprop is reverse accumulation.

Guided checkpoint

Match mode to efficiency regime.

Source: Sec. 5.6, pp. 159-164

Higher-order derivatives

Curvature distinguishes minima, maxima, and saddles locally.

Guided checkpoint

For twice differentiable scalar $f$ .

Source: Sec. 5.7, pp. 164-165

Second-order Taylor model

Taylor expansion combines local value, slope, and curvature.

Guided checkpoint

Which terms appear around $x_0$ ?

Source: Sec. 5.8, pp. 165-170

Section 04

Linearization to Exercise-style chain rule

Linearization

f(3)+f'(3)(.1)=9+6(.1)=9.6

Guided checkpoint

For $f(x)=x^2$ , linearize at $x_0=3$ and evaluate the approximation at $3.1$ .

Source: Sec. 5.8, pp. 165-170

Further-reading bridge

The chapter points from calculus identities to AD systems.

Guided checkpoint

Which topic extends derivatives to parameterized programs at scale?

Source: Sec. 5.9, p. 170

Exercise-style chain rule

Chain rule gives

3(2x+1)^2 \times 2=6(3)^2=54

Guided checkpoint

Let $z=(2x+1)^3$ . Compute $dz/dx$ at $x=1$ .

Source: Chapter 5 exercises, pp. 170-171

Knowledge check

Turn understanding into recall.

The quiz now follows the same concepts in scored form. You can return to this lesson from the quiz whenever a gap appears.

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