## Related questions with answers

Verify that

$w _ { x y } = w _ { y x }.$

w = ln (2x + 3y)

Solutions

VerifiedIn this task we are in essence asked to verify that the mixed partial derivatives theorem holds for the given function $w(x,y)$. The theorem states that if both first and second order partial derivatives are continuous, then the mixed partial derivatives $w_{xy}$ and $w_{yx}$ are the same. The procedure we will use here is straightforward, we will first find $w_{yx}$ and then we will find $w_{xy}$. After finding both expressions, we will compare them to verify the result.

$\dfrac{\partial w}{\partial x}=\dfrac{\partial}{\partial x}\left[\ln(2x+3y)\right]$

$=\dfrac{1}{2x+3y}\cdot\dfrac{\partial}{\partial x}\left(2x+3y \right)$

$=\dfrac{2}{2x+3y}$

$\dfrac{\partial w}{\partial y}=\dfrac{\partial}{\partial y}\left[\ln(2x+3y)\right]$

$=\dfrac{1}{2x+3y}\cdot\dfrac{\partial}{\partial y}\left(2x+3y \right)$

$=\dfrac{3}{2x+3y}$

Compute all first partial derivatives.

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