Solving a Chain Rule Problem with F(x,y)

In summary, the conversation is about solving a chain rule problem involving the function F(x,y)=f((x-y)/(x+y)). The problem provides values for f' and f'' at certain points, and the goal is to find the partial derivatives of F at (2,-1), as well as the second order partial derivative. The final answers for the partial derivatives should be -80 and 372, and the person is interested in the method used to solve the problem. They mention an approach using t=x-y and s=x+y, but got stuck.
  • #1
Yankel
395
0
Hello all,

I need some help with this chain rule problem.

\[F(x,y)=f\left (\frac{x-y}{x+y} \right )\]

It is known that:

f'(1)=20,f'(2)=30, f'(3)=40

and

\[f''(1)=5,f''(2)=6,f''(3))=7\]Find

\frac{\partial F}{\partial x}(2,-1)

and

\[\frac{\partial^2 F}{\partial x\partial y}\]The final answers should be -80 and 372.

I am less bothered with the final numbers (although would like to get there). It is the way that I am interested in. I was thinking to set t=x-y and s=x+y, but it got me stuck.
 
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  • #2
Are both derivatives being evaluated at $(2,-1)$?
 

FAQ: Solving a Chain Rule Problem with F(x,y)

1) What is the chain rule and how does it apply to solving a problem with F(x,y)?

The chain rule is a mathematical concept used to find the derivative of a composite function. In the context of solving a problem with F(x,y), the chain rule helps us find the derivative of a function that is composed of two or more functions.

2) What are the steps for solving a chain rule problem with F(x,y)?

The steps for solving a chain rule problem with F(x,y) are as follows: first, identify the outer function (F) and the inner function (f). Then, find the derivative of the outer function (F') and the derivative of the inner function (f'). Finally, substitute these values into the chain rule formula: F'(x,y) = F'(f(x,y)) * f'(x,y).

3) Can the chain rule be applied to any function with multiple variables?

Yes, the chain rule can be applied to any function with multiple variables as long as it is a composite function. This means that the function is composed of two or more functions.

4) How does the chain rule help us find the rate of change in a multivariable function?

The chain rule helps us find the rate of change in a multivariable function by allowing us to break down the function into smaller, more manageable parts. This allows us to find the rate of change for each individual part and then combine them using the chain rule formula to find the overall rate of change.

5) Can the chain rule be generalized to solve problems with more than two functions?

Yes, the chain rule can be generalized to solve problems with any number of functions. The formula for the chain rule can be extended to include as many functions as needed, as long as they are composed together. However, the calculation process may become more complex and time-consuming as the number of functions increases.

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