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If $f(x,y)$ is a function, to compute $\partial f / \partial y$ at some point $(a,b)$, is it always acceptable to plug in the value $a$ for $x$ before computing the derivative? I am convinced the answer is "yes", and that no justification needs to be given: literally the definition of the partial derivative at a point $(a,b)$ involves doing this. Is this correct (including the justification)?

The source of my thinking about this question is Problem 2-19 in Spivak's Calculus on Manifolds, which asks us to compute $\partial f / \partial y$ at a point of the form $(1,y)$, where $f$ is some more complicated version of $$f(x,y) = x^{x^y}.$$ Clearly the trick is to plug in $x = 1$ first, and I am convinced no justification for this step needs to be given, but something about it makes me nervous.

Thank you!

CJD
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Yes, this is correct. By definition $\frac{\partial f}{\partial y}|_{(a,b)}$ is the derivative of the function $y\mapsto f(a,y)$ evaluated at $y=b$.

K.Power
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  • Thank you! I will mark this as the accepted answer after waiting a few hours. – CJD Apr 09 '19 at 19:12
  • Anytime. But please mark a better answer as the accepted one if someone else puts more work into the explanation. – K.Power Apr 09 '19 at 19:15