Review of functions of two variables: Chain rule and implicit differentiation of a function of two or more variables

 In this post, we are going to review the chain rule for the partial derivatives of z = f(x,y) where x and y are functions of two independent variables u and v. Then we will review the higher order partial derivatives.

Chain rule of a function of two variables x and y where x and y are function of two independent variables

Implicit differentiation of a function of more than two variables

Next time will review Directional derivatives, maxima and minima.

Review of a function of two variables: Chain rule

 The Chain rule

• The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables.
• Three diagrams are useful for deriving formulas for functions of more than one variable, where each independent variable depends also of the other variables.

Chain rule of a function of two variables for one independent variable

Review of function 2 variables ( derivatives)

 Partial derivatives

Tangent plane and linear approximation

Review the following posts:

Derivatives of a function of two variables

Calculating partial derivatives of a function of two variables without using limit

Partial derivative value using a contour map

Tangent plane to a surface at a point

Linear approximation for function of two variables

Notion of differential of a function of two variables

Review of a function of two variables (continued)

 Objectives:

1) Review level curves and contour maps

2) Limit and continuity

Level curves and contour map

Given a function f(x,y) and a number c of its range, a level curve of the function with two variables f(x, y) for the value c is the set of points that satisfy the equation f(x, y) = c. A graph of the various level curves of a function with two variables is called a contour map.

Review the corresponding posts:

Level curves

Level curves. Making a contour map (continued)

Limit and Continuity of a function of two variables

Review the following post:

Limit of a function of two variables

Limit of a function of two variables at a boundary point

Continuity of a function of two variables

Operations on the continuity of a function of two variables

Review of functions of several variables

  Functions of several variables

I finish the chapter concerning the function of several variables and start today with a review. In this post, I review the graph of a function of 2 variables.

A. Function of 2 variables

1) Graph

a) The graph of a function of 2 variables in R³ is called a surface. It can be studied using level curves and vertical traces. A set of level curves is called a contour map.

Here are the post that refers to the graph of a function of 2 variables:

Graph of a function of 2 variables

Graph of a function of 2 variables (continued)

(to be continued)

Higher order partial derivatives of a function of two variables

 Objective: Find all second order partial derivatives of a function of two variables

High order partial derivatives

Let's consider a function of two variables. We can calculate the partial derivatives of this function. If we calculate the partial derivatives of the first partial derivatives, we obtain second order partial derivatives. If we calculate the partial derivatives of these second order partial derivatives, we obtain third order partial  derivatives and so on.

Let's consider the function:

Its partial derivatives are:

The partial derivatives of these partial derivatives are:

𝛿²f/δx² = 8x and δ²f/δy² = -8x + 30y. These new partial derivatives are called second order partial derivatives.

There are 4 second order partial derivatives of any function provided they all exist

Example

Solution

Practice

Partial derivatives of a function of three variables

 The partial derivative of a function of three variables can be calculated the same way we calculate the partial derivatives of a function of two variables. To calculate the partial derivative with respect to x, we consider y and z constants then we calculate the derivative considering only x as a variable. In general to calculate the partial derivative to one variable, we calculate the derivative with respect the considered variable while we keep the other two variables constant.

Definition

Example

Solution

To  calculate the partial derivative with respect to x using the limit definition, let's start by using the formula                                                                                                                                              

Let's start by calculating f(x + h, y, z):                                                                                                         

Let's substitute f(x+h, y, z) and f(x,y,z) in the formula:                                                                                   

                                                                                                     

Practice