Problem 39
Question
Verify that \(f_{x y}=f_{y x}\) for the following functions. $$f(x, y)=e^{x+y}$$
Step-by-Step Solution
Verified Answer
Question: Verify if \(f_{xy} = f_{yx}\) for the function \(f(x, y) = e^{x + y}\).
Answer: Yes, for the given function \(f(x, y) = e^{x + y}\), \(f_{xy} = f_{yx} = e^{x + y}\).
1Step 1: 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:
$$
f_x(x, y) = e^{x + y}
$$
2Step 2: 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:
$$
f_{xy}(x, y) = e^{x + y}
$$
3Step 3: 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:
$$
f_y(x, y) = e^{x + y}
$$
4Step 4: 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:
$$
f_{yx}(x, y) = e^{x + y}
$$
5Step 5: 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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