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I have to find the $\partial^2 z/\partial x \partial y$ of $z=e^{xy}$. I know how to find $\partial^2 z/\partial x^2$ which by the way is $y^2 e^{xy}$ but not this one...can you give me a little hint? and please tell me how to find this type of "power" partial derivatives,like $\partial^3z/\partial x^2 \partial y$..

mwoua
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user77207
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1 Answers1

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Hint : $$\frac{\partial^2z}{\partial x \partial y} = \frac{\partial}{\partial x} \left(\frac{\partial z}{\partial y}\right)$$

For all you others "types of power partial derivatives" as you say, simply split them.

Example : $$\frac{\partial^n z}{\partial x_1 \partial x_2 \partial x_3 \dots \partial x_n} = \frac{\partial}{\partial x_1}\left(\frac{\partial}{\partial x_2}\left(\frac{\partial}{\partial x_3}\left(\dots\left(\frac{\partial z}{\partial x_n}\right)\right)\right)\right) $$

mwoua
  • 865