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These questions are related but don't answer my specific question: 1, 2, 3.

Assume we have a two-variable function where the second depends on the first: $z=f(x,y(x))$

Is it possible in this case to calculate the integral of the partial derivative of $f$ with respect to $x$, that is independent of the specific form of $f$ and $y(x)$?

$$\int\frac {\partial f}{\partial x}(x,y(x))dx=?$$

We cannot apply the fundamental theorem of calculus directly as we would if it was not a partial but a standard derivative:

$$\int\frac {d}{dx}f(x,y(x))dx=f(x,y(x))+C$$

Is there some similar (general!) formula for the partial derivative case?

user56834
  • 12,323
  • The notation is unclear. Do you mean $\frac{\partial f}{\partial x}(x, g(x))$, i.e., take the partial derivative of $f(x,y)$ with respect to $x$, and then substitute some function $y=g(x)$ afterwards? In that case: no, there is no general rule. (How could there be? The derivative has no idea what function $g(x)$ you're going to plug in afterwards...) – Hans Lundmark Aug 31 '17 at 08:34
  • Yes that's what I meant. Isn't that exactly what that notation means? – user56834 Aug 31 '17 at 08:41
  • Well, when you write $\frac{\partial}{\partial x} \text{something}$, it means take the partial derivative of the whole expression “something”. And in your case, that expression is $f(x,y(x))$, which depends only on one variable $x$. So there would be no difference between $\frac{\partial}{\partial x}$ and $\frac{d}{dx}$! – Hans Lundmark Aug 31 '17 at 08:46
  • Writing $\frac{\partial f}{\partial x}$ is different, since then the horizontal line acts like brackets, saying “differentiate the function $f$” (meaning $f(x,y)$ with independent variables $x$ and $y$) first. – Hans Lundmark Aug 31 '17 at 08:48
  • Ah that makes sense. I've changed it, and I suppose this notation makes the most sense? – user56834 Aug 31 '17 at 09:14
  • Yes, now it's clear. – Hans Lundmark Aug 31 '17 at 09:15
  • Yes it works $$\int \frac{\partial f(x,y(x))}{\partial x} , dx=f(x,y(x))+C$$ – Raffaele Aug 31 '17 at 15:18

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