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Given a function $f(x,y,t)$, is it correct to say

$$\frac{d f}{d x} = \frac{\partial f}{\partial x} \text{ ?}$$

Michael Hardy
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mononono
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2 Answers2

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The function $f$ is of three variables, so you should be using $$\frac{\partial f}{\partial x},$$ or $f_x$ or $f_1$ to denote the partial derivative of $f$ with respect to $x$.

Here $df/dx$ has no meaning because $f$ is a function of three variables. You would only use this notation (or $f'$) if $f$ was a function of $x$ alone. (You might also use $f'$ to denote the Jacobian matrix of partial derivatives for a function of many variables.)

smcc
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  • Agreed except for the last bit. If $x$, $y$, and $z$ were all functions of $t$, so that we have $f(x(t), y(t), z(t))$, then it would make sense to say $\dfrac{df}{dt}$. +1 –  Jul 22 '16 at 15:38
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    Yes, but it is really abuse of notation to do so. You should really write $g(t)=f(x(t),y(t),z(t))$ and then denote the derivative as $g'(t)$ or $dg(t)/dt$. – smcc Jul 22 '16 at 15:40
  • To see why it is bad notation, notice that if we have $f(x(t),t)$, and use the notation $df/dt$ for the derivative of this composition, we get: $\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial t}$, so on the LHS $f$ is a function of one variable, while on the RHS it is a function of two variables. – smcc Jul 22 '16 at 15:46
  • I've never been taught that but I do see how a new $g(t)$ is simpler. I don't agree with the LHS vs. RHS number of variables thing. –  Jul 22 '16 at 15:49
  • I think the whole "total derivative" thing is silly/confusing terminology. The total derivative they write about there is just the partial derivative (or derivative) of a composition of functions. – smcc Jul 22 '16 at 15:51
  • But that's what $f(x(t), t)$ and $f(x(t), y(t), z(t))$ are - compositions of functions. The example at the top of the article is almost the same as yours: $f(t, x, y)$ vs. $f(x(t), t)$. –  Jul 22 '16 at 16:00
  • Yeah, but why call it a total derivative as if it is something different to the usual derivative or partial derivative? The example at the top of the article is just calculating the derivative of $g(t)=f(t,x(t),y(t))$. – smcc Jul 22 '16 at 16:05
  • I don't know the history of the name but it's probably at least partly to keep things separate. They are a little different. It makes sense to say "derivative of $g$" because $g$ is a function of one variable. It doesn't make sense to just say "derivative of $f$" because $f$ is a function of multiple variables. Even though each of those variables are themselves functions of $t$, $f$ is still defined in terms of multiple variables. –  Jul 22 '16 at 16:27
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For a ternary function $f(x,y,t)$, the expression $\dfrac{df}{dz}$ would usually be read as the total derivative of $f$ with respect to the exogenous argument $z$. In general, you have for this total derivative that $$ \frac{df}{dz} = \frac{\partial f}{\partial x} \cdot \frac{dx}{dz} + \frac{\partial f}{\partial y} \cdot \frac{dy}{dz} + \frac{\partial f}{\partial t} \cdot \frac{dt}{dz}. $$

For $z=t$, if neither $x$ nor $y$ depends on $t$, then $\frac{dx}{dt} = \frac{dy}{dt} = 0$, and since $\frac{dt}{dt} = 1$ you indeed get that $$ \frac{df}{dt} = \frac{\partial f}{\partial t}. $$

fgp
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