CHAPTER 05 / 40
Calculus
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Gradients, Jacobians and the chain rule are the literal engine of backpropagation — every weight update in every neural network ultimately calls these tools.
- backprop
- autograd
- neural training
- Step 25~20m25⚡Challenge
Differentiation of Univariate Functions
The derivative of a function measures its instantaneous rate of change at . Formally, . Geometrically, it is the slope of the tangent line t…
Open lesson - Step 26~20m3D · 125★Most cinematic⚡Challenge
Partial Derivatives and Gradients
For a function of several variables, the partial derivative is the rate of change of as varies while all other variables are held fixed. Com…
Open lesson - Step 27~15m15⚡Challenge
Gradients of Vector-Valued Functions
When maps vectors to vectors, the analog of the gradient is a matrix called the Jacobian, . Its entry is — the sensitivity of the -th output…
Open lesson - Step 28~15m15⚡Challenge
Gradients of Matrices
In ML, loss functions often depend on whole matrices of parameters (think weight matrices in a neural net layer). The gradient with respect…
Open lesson - Step 29~20m3D · 125★Most cinematic⚡Challenge
Backpropagation and Automatic Differentiation
Backpropagation is the chain rule, applied efficiently to a computational graph — a directed acyclic graph where each node is a primitive op…
Open lesson - Step 30~20m25⚡Challenge
Higher-Order Derivatives
Taking the derivative of a derivative gives higher-order derivatives. In 1D, measures curvature — how fast the slope is itself changing. Lar…
Open lesson - Step 31~20m35⚡Challenge
Linearization and Taylor Series
A Taylor series approximates a smooth function by a polynomial. Near a point , the -th order approximation is . The first-order approximatio…
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