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Chapter 13: Problem 39

Verify that \(f_{x y}=f_{y x}\) for the following functions. $$f(x, y)=e^{x+y}$$

Short Answer

Expert verified
Question: Verify if \(f_{xy} = f_{yx}\) for the function \(f(x, y) = e^{x + y}\). Answer: Yes, for the given function \(f(x, y) = e^{x + y}\), \(f_{xy} = f_{yx} = e^{x + y}\).

Step by step solution

01

Differentiate partially concerning x

To find the partial derivative of \(f(x, y) = e^{x + y}\) concerning x, use the chain rule and differentiate the power e: $$ f_x(x, y) = e^{x + y} $$
02

Differentiate partially concerning y

To find the partial derivative of \(f_x(x, y) = e^{x + y}\) concerning y, use the chain rule and differentiate the power e: $$ f_{xy}(x, y) = e^{x + y} $$
03

Differentiate partially concerning y

Now, we need to differentiate the original function partially concerning y. Again, apply the chain rule and differentiate the power e: $$ f_y(x, y) = e^{x + y} $$
04

Differentiate partially concerning x

Finally, to find the partial derivative of \(f_y(x, y) = e^{x + y}\) concerning x, use the chain rule and differentiate the power e: $$ f_{yx}(x, y) = e^{x + y} $$
05

Compare the results

Now that we have found \(f_{xy}(x, y)\) and \(f_{yx}(x, y)\), let's compare them to verify if they are equal: $$ f_{xy}(x, y) = e^{x + y} = f_{yx}(x, y) $$ Since \(f_{xy}(x, y) = f_{yx}(x, y)\), we have successfully verified that \(f_{xy} = f_{yx}\) for the given function \(f(x, y) = e^{x + y}\).

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