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Lecture# Partial Derivatives

Description

This lecture covers the concept of partial derivatives, focusing on the computation and interpretation of these derivatives in various contexts. The instructor explains how to calculate partial derivatives and discusses their applications in optimization and tangent plane approximation.

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In course

Instructor

MATH-431: Theory of stochastic calculus

Introduction to the mathematical theory of stochastic calculus: construction of stochastic Ito integral, proof of Ito formula, introduction to stochastic differential equations, Girsanov theorem and F

Related concepts (66)

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function with respect to the variable is variously denoted by It can be thought of as the rate of change of the function in the -direction.

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Derivative

In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.

Total derivative

In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as considering all partial derivatives simultaneously. The term "total derivative" is primarily used when f is a function of several variables, because when f is a function of a single variable, the total derivative is the same as the ordinary derivative of the function.

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Related lectures (156)

Partial Derivatives: Understanding and ApplicationsMATH-105(a): Advanced analysis II

Explores the computation and significance of partial derivatives in determining rates of change.

Partial DerivativesMATH-431: Theory of stochastic calculus

Covers the concept of partial derivatives and properties of saddle points.

Derivatives and Tangent Planes

Covers derivatives, differentiability, and tangent planes for functions of one and two variables.

Partial Derivatives and FunctionsMATH-106(f): Analysis II

Explores partial derivatives and functions in multivariable calculus, emphasizing their importance and practical applications.

Partial Derivatives: Higher Order and Applications

Covers the concept of partial derivatives of higher order and their applications in various contexts.