# Multivariable calculus partial derivatives/chain rule

• kaitamasaki
In summary, the problem involves using the chain rule from multivariable calculus to find the partial derivatives of F at specific points, and then using those values to calculate the partial derivatives of W. The given numbers represent the values of u, v, s, and t at which the derivatives need to be evaluated.
kaitamasaki

## Homework Statement

[PLAIN]http://img195.imageshack.us/img195/8196/mathqshwk4.jpg
Find [URL]https://webwork.elearning.ubc.ca/webwork2_files/tmp/equations/f3/d5a53499bbf12cafa144440e3095781.png[/URL]
and [URL]https://webwork.elearning.ubc.ca/webwork2_files/tmp/equations/b1/ce11cb4b480b6159791b1605b6f2681.png[/URL]

## Homework Equations

Chain rule from multivariable calculus.

## The Attempt at a Solution

I have tried doing the obvious: multiplying the given numbers, but I am just confused by what F_u and F_v at 5,3 means.

Instinct tells me the question requires F_u at (1,0), multiplied by u_s at (1,0), but none of the answers I've tried have worked. I am more confused by the way this question is asked because I understand multivariable chain rule quite well already to do most other questions.

Last edited by a moderator:
$F_u(5,3)$ means means the partial derivative of F, with respect to u, evaluated at u= 5, v= 3. That's pretty much standard notation. When s= 1, t= 0, then u= 5, v= 3 and you evaluate F and its derivatives at u and v, not s and t. That's why they are labeling F(u(x,t), v(s,t)) as "W(s, t)" rather than "F(s,t)".

From the chain rule, directly,
$$W_s(1, 0)= F_u(5, 3)u_s(1, 0)+ F_v(5, 3)v_s(1, 0)$$
and
[tex]W_t(1, 0)= F_u(5, 3)u_t(1, 0)+ F_v(5, 3)v_t(1, 0)[/itex]

and all of those numbers are given.

## What is multivariable calculus?

Multivariable calculus is a branch of mathematics that deals with functions of more than one variable. It extends the concepts and techniques of calculus to functions with multiple independent variables.

## What are partial derivatives?

Partial derivatives are the derivatives of multivariable functions with respect to one of its variables while keeping the other variables constant. They represent the rate of change of a function in a specific direction.

## What is the chain rule in multivariable calculus?

The chain rule in multivariable calculus is a method for finding the derivative of a composition of two or more functions. It states that the derivative of a composite function is the product of the derivatives of the individual functions.

## Why is the chain rule important in multivariable calculus?

The chain rule is important in multivariable calculus because many functions are composed of multiple functions, and it allows us to find the derivative of complicated functions. It is also essential in applications such as optimization and curve sketching.

## How do you calculate partial derivatives using the chain rule?

To calculate partial derivatives using the chain rule, we first find the derivatives of the individual functions. Then, we substitute the variables with the corresponding functions and multiply the derivatives. Finally, we evaluate the resulting expression at the desired point.

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