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Can someone tell me the correct partial derivative of $x^{y^z}$ on the variable z. I got two solutions:

$x^{y^z}\ln\left(x^y\right)$

and

$x^{y^z}y^z\ln x\ln y$

They both seem right to me so I am confused. Which one is correct and why?

Davide Giraudo
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mandm
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    How do you have the order of operations? Is it $x^{(y^z)}$ or $(x^y)^z$? – Simply Beautiful Art Dec 27 '16 at 13:13
  • @SimpleArt I've never seen nested exponents meant to be evaluated from left to right. Especially when you have to write x^{y^z} to make it render correctly, I would think there is little to no ambiguity. If they had written "x^y^z", on the other hand... – Arthur Dec 27 '16 at 13:19
  • It doesn't say anything in the task, only $x^{y^z}$ and that is how I got these two solutions, so the problem is only that the task is not precisly defined? – mandm Dec 27 '16 at 13:19
  • @Arthur Well, by the looks of it, the first solution the OP got was the second case I gave, so there does seem to be ambiguity. – Simply Beautiful Art Dec 27 '16 at 13:21
  • @SimpleArt Ahh, I didn't see that that was how he got the two different answers. As hinted to earlier, it didn't even occur to me to interpret the exponents in that direction. Nice catch. – Arthur Dec 27 '16 at 13:22
  • The task is precisely defined without any mathematical ambiguity. The expression $x^{y^z}$ means only $x^{(y^z)}$. It is a common confusion, though, so I admit that there's a perceived ambiguity there. I think part of learning experience to take away from this question is to learn how the notation for nested exponents actually works. – zipirovich Dec 27 '16 at 15:42

2 Answers2

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If $$f(x,y,z) = x^{y^{z}} \text{ then } \frac{\partial}{\partial z} f(x,y,z) = \log(x) y^z \log y x^{y^{z}}$$ And if, $$f(x,y,z) = (x^y)^{z} \text{ then } \frac{\partial}{\partial z} f(x,y,z) = (x^y)^{z} \log(x^y)$$ These results are different due to the different ways the power of $z$ has been executed. Hope it helps.

Simply Beautiful Art
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If we have $$x^{y^z}$$ then we get by the chain rule $${x}^{{y}^{z}}{y}^{z}\ln \left( y \right) \ln \left( x \right) $$

Dr. Sonnhard Graubner
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