Groups
Category
Level
Implicit differentiation lets you find slopes and higher derivatives even when y is given indirectly by an equation F(x,y)=0.
Taylor series approximate a complicated function near a point by a simple polynomial built from its derivatives.
The Hessian matrix collects all second-order partial derivatives of a scalar function and measures local curvature.
The Jacobian matrix collects all first-order partial derivatives of a vector-valued function, describing how small input changes linearly affect each output component.
The multivariable chain rule explains how rates of change pass through a pipeline of functions by multiplying the right derivatives (Jacobians) in the right order.
The gradient \(\nabla f\) points in the direction of steepest increase of a scalar field and its length equals the maximum rate of increase.
Partial derivatives measure how a multivariable function changes when you wiggle just one input while keeping the others fixed.
Derivatives measure how fast a function changes, and rules like the product, quotient, and chain rule let us differentiate complex expressions efficiently.
A limit describes what value a function approaches as the input gets close to some point, even if the function is not defined there.