Is there a function f : R^2 \to R whose partial derivatives are \frac {\delta f}{\delta x} = x +...


Is there a function {eq}f : R^2 \to R {/eq} whose partial derivatives are {eq}\frac {\delta f}{\delta x} {/eq}= x + 4y and {eq}\frac {\delta f}{\delta y} {/eq}= 3x - y? If so, find this function and if not, explain why it can't exist.

Mixed Partial Derivatives:

For a real valued twice continuously differentiable function, we have:

{eq}\displaystyle \frac{\partial ^2f}{\partial x\partial y}=\frac{\partial ^2f}{\partial y\partial x}\\ {/eq}

We will use this fact to solve this problem.

Answer and Explanation: 1

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If such a function f exists, then we would have:

{eq}\displaystyle \frac{\partial ^2f}{\partial x\partial y}=\frac{\partial ^2f}{\partial y\partial...

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Learn more about this topic:

The Chain Rule for Partial Derivatives


Chapter 14 / Lesson 4

This lesson defines the chain rule. It goes on to explore the chain rule with partial derivatives and integrals of partial derivatives.

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