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

 Goal: Calculate the derivation of a function of two variables x and y where x and y are functions of two independent variables.

Let's recall earlier when x and y are function of one variable, the formula is a sum of 2 products of derivatives. where z is differentiated partially with respect to x and y. Then x and y are differentiated normally with respect to the independent variable. To recall the formula easily, let's start by writing the first term of each of the 2 products. These first terms are respectively the partial derivatives of z with respect to x and with respect to y. Let's start as follows;

dz/dx = ẟz/ẟx ..... + ẟz/ẟy......

Now complete the first and the second product. The term that completes the first product is the regular derivative of x with respect to the independent variable (t). The term that completes the second product is the regular derivative of y with respect to the independent variable (t). Then we have:

dz/dx = ẟz/ẟx.dx/dt + ẟz/ẟy.dy/dt

Chain rule for a function of two independent variables

The expressions in these formulas are going to be in partial derivatives only.. We have a function of two variables z = f(x, y) where x and y are functions of two independent variables: x = x(u,v) and y = (u, v). We are going to have 2 expressions of partial derivatives. One expression of partial derivative  of z with respect to u and  another expression with respect to v. In the first expression we derive with respect to u and in the second we derive with respect to v. Each expression of partial derivative is a sum of products of derivatives as for the derivative for one variable.

Let's start with the partial derivative with respect to u. In this expression we derive x and y partially with respect to to u. Let's start by writing the first term of each product: they are partial derivative of z with respect to x and y.  We have:

 ẟz/ẟu =  ẟz/ẟx ....+   ẟz/ẟy....

Let's complete each product by deriving x partially with respect  to u in the first product and y partially with respect to u in the second product

ẟz/ẟu =  ẟz/ẟx.ẟx/ẟu +   ẟz/ẟy.ẟyẟu.

Let's write the expression of ẟz/ẟv  . We follow the same procedure by writing the first terms of each product.

 ẟz/ẟv =  ẟz/ẟx ....+   ẟz/ẟy....

Let's complete the second terms of the products by differentiating x and y partially with respect to v.

ẟz/ẟu =  ẟz/ẟx.ẟx/ẟv. +   ẟz/ẟy.ẟx/ẟv.

Theorem. Chain rule for two independent variables

Suppose x = (u, v) and y = (u, v) are differentiable functions of u and v and z = f(x, y) is a differentiable function of x and y. Then z = f((u,v), (u,v)) is differentiable of u and v. We have the following formulas:

and

Example

Solution

Let's apply the first formula:

 

Let's calculate the partial derivatives:

Let's substitute everything in the formula. We have:

Let's substitute x and u:

Let's now use the second formula, First let's calculate ẟx/ẟv and ẟ/ẟv:

We need to calculate the missing terms ẟx/ẟv and ẟ/ẟv:

Let's substitute the partial derivatives:

Let's substitute x and y:

Practice

and