Solved: "Find the mixed Partial Derivatives for the following..."

f480forever2rz

Answered question

2021-11-17

Find the mixed Partial Derivatives for the following functions

w = e^{x} + x \ln (y) + y \ln (x)

Florence Evans

Beginner2021-11-18Added 16 answers

Step 1
The given function is:

w (x, y) = e^{x} + x \ln (y) + y \ln (x)

Step 2
Now:

\frac{\partial w}{\partial x} = e^{x} + \ln (y) + \frac{y}{x}

\frac{\partial}{\partial y} (\frac{\partial w}{\partial x}) = 0 + \frac{1}{y} + \frac{1}{x}

\frac{\partial^{2} w}{\partial y \partial x} = \frac{1}{y} + \frac{1}{x}

Therefore:

w_{y x} = \frac{1}{y} + \frac{1}{x}

And

\frac{\partial w}{\partial y} = 0 + \frac{x}{y} + \ln (x)

\frac{\partial}{\partial x} (\frac{\partial w}{\partial y}) = \frac{1}{y} + \frac{1}{x}

\frac{\partial^{2} w}{\partial x \partial y} = \frac{1}{y} + \frac{1}{x}

Therefore:

w_{x y} = \frac{1}{y} + \frac{1}{x}

Hence the mixed partial derivatives are:

w_{x y} = \frac{1}{y} + \frac{1}{x} and w_{y x} = \frac{1}{y} + \frac{1}{x}

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