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More specifically, I see $\frac{\partial}{\partial x}f(x,g(x))$ as meaning take the derivative of $f(x,g(x))$ w.r.t $x$.

On the other hand, I see $f_x(x,g(x))$ as meaning the partial derivative of $f(x,y)$ evaluated at $(x,g(x))$

Do these notations have clearly defined meanings -- i.e. is it clear whether we take the derivative and then evaluate or evaluate and then take the derivative -- or does it depend on context?

If they do not, how can I be clear in which order to take the derivative and evaluate?

As an illustration, consider $f(x,y) = x^2+y^2$

Then $f_x(x,y) = 2x$

I would then say that $f_x(x,g(x))= 2x$

Whereas I would say that $\frac{\partial}{\partial x}f(x,g(x)) = \frac{\partial}{\partial x}\left(x^2 + g(x)^2 \right)= 2x + 2g(x)g'(x) = f_x(x,g(x) )+f_y(g(x))g'(x)$

user106860
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1 Answers1

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You have to distinguish between $\frac d {dx}$ and $\frac {\partial} {\partial x}$. Your calculation of $\frac {\partial} {\partial x} f (x,g(x))$ is wrong and you have to intepret it as $f_x$ evaluated at $(x,g(x))$.

Kavi Rama Murthy
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  • How is my calculation for $\frac{\partial}{\partial x}(f(x,(g(x)))$ wrong? And how is the distinguish between $\frac{d}{dx}$ and the partial form relevant here, given that $f(x,g(x))$ is only a function of $x$, so $\frac{d}{dx}$ and $\frac{\partial}{\partial x}$ would give the same thing? – user106860 Mar 22 '19 at 00:09
  • It is a question of conventions. In the expression $\frac {\partial} {\partial x} f (x,g(x))$ you are suppose to club $\frac {\partial} {\partial x}$ and $f$ together and $(x,g(x))$ comes later. – Kavi Rama Murthy Mar 22 '19 at 00:18
  • Is there a way to say $\frac{\partial}{\partial x}$ of $f(x,g(x))$? Maybe define $z(x)=f(x,g(x))$ and write $\frac{\partial}{\partial x} z(x)$? – user106860 Mar 25 '19 at 01:58
  • @user106860 For the derivative of $f(x,g(x))$ w.r.t. $x$ we always use $\frac d {dx}$ and do not use the symbols $\partial$. – Kavi Rama Murthy Mar 25 '19 at 05:18