Calculus
Key points
Partial derivatives
When a function has more than one input, you differentiate with respect to one variable at a time and treat the others as if they were fixed numbers. Differentiating f(x,y) with respect to x while holding y still gives the partial derivative ∂f/∂x. Collect all the partials into one vector and you get the gradient, ∇f = (∂f/∂x, ∂f/∂y), which points in the direction the function climbs fastest.
A worked partial derivative
Take f(x,y) = x²y + 3y². To get ∂f/∂x, treat y as a constant: x²y differentiates to 2xy, and 3y² is a constant so it vanishes. To get ∂f/∂y, treat x as a constant instead: x²y differentiates to x², and 3y² gives 6y. So ∂f/∂x = 2xy and ∂f/∂y = x² + 6y.
Taylor series
A Taylor series rewrites a function as a polynomial that matches it near a chosen point x = a — the more terms you keep, the wider the region where the match is good. For example, sin x ≈ x − x³/6 + x⁵/120 near zero. This is how a computer actually evaluates sine, cosine, and logarithms: it never "knows" them directly, it adds up a few polynomial terms.
Double integrals
A double integral ∬f(x,y) dx dy adds up a two-variable function across a whole region of the plane, which is how you find a volume, a total probability, or the mass of a flat object. You evaluate it as two ordinary integrals in a row: integrate with respect to one variable first, then integrate the result with respect to the other.
Gradient descent (machine learning)
Gradient descent repeatedly nudges the parameter w a small step in the direction opposite the gradient of the loss L(w): w := w − η · ∂L/∂w. The gradient points uphill, so moving against it moves the loss down. The step size η is the learning rate — too small and training crawls, too large and it overshoots.
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