1

Let's have the following problem:

$$\frac{d \left[ \left(\frac{x_1}{x_2} \right)^\sigma + x_1 \right]}{d \frac{x_1}{x_2}}$$

I want to differentiate the function by the fraction and I was told in previous questions that this can be solved by substitution such as this:

$$ z = \frac{x_1}{x_2} \qquad ; \qquad x_1(z) = z\cdot x_2$$

$$\frac{d \left[ \left( z \right)^\sigma + x_1(z) \right]}{d z}$$

However, previously I was using only partial derivation but now I would like to this with the use of total derivation.

Can I still use the same approach and substitute while treating the $x_2$ as a constant?

$$\frac{d \left[ \left( z \right)^\sigma + x_1(z) \right]}{d z} = \sigma z^{\sigma - 1} + \frac{d z \cdot x_1}{d z} = \sigma z^{\sigma - 1} + x_2$$

Or is this approach invalid and $x_2$ should be considered again as a function of $z$?

Does this mean that the correct result would be:

$$\frac{d \left[ \left( z \right)^\sigma + x_1(z) \right]}{d z}= \sigma \left( \frac{x_1}{x_2} \right)^{\sigma - 1} + \frac{d x_1(z)}{d \frac{x_1}{x_2}} = \sigma \left( \frac{x_1}{x_2} \right)^{\sigma - 1} + dx_1$$

Where $d x_1$ denotes by how much the $x_1$ changes? What would be the interpretation of it?

Athaeneus
  • 191
  • You mean total differential or what? If yes, what was your approach? The equation begins with $dz=...$. – callculus42 Apr 01 '23 at 02:54
  • First ask yourself what does $dx_1/d(x_1/x_2)$ as a "total derivative" mean? Intuitively if a ratio changes, can that tell you how the numerator changes? – Golden_Ratio Apr 01 '23 at 07:18
  • What I mean by that is the difference between $d$ and $\partial$. As I understand this, the $d$ implies that if some variable changes, changes in other variables which would be affected by this are allowed as well. So I think that $d x_1 / d(x_1/x_2)$ means that we allow the $x_1$ to change as the ratio changes. What concetns total derivatives, @callculus42, I follow this post on it: https://math.stackexchange.com/a/174272/1116386 – Athaeneus Apr 01 '23 at 07:47
  • 1
    @Athaeneus In the link you cite, the total derivative is taken wrt $\theta$ where $x,y$ depend on $\theta$. In contrast, in this case you wish to differentiate with respect to $x_1/x_2$ but you have not specified how $x_1$ depends upon $x_1/x_2$ (or how $x_2$ depends upon $x_1$). A change in the ratio $x_1/x_2$ tells you nothing about how $x_1$ changes: do you see why? – Golden_Ratio Apr 04 '23 at 17:09
  • @Golden_Ratio Thank you very much! I think I am getting closer. Just to be sure, do I understand it correctly, that $x_1/x_2$ might change because either $x_1$ has changed, $x_2$ has changed, or both? What happens then if I do the substitution? If I say $x_1(z) = x_2 z$, does this specify, how the $x_1$ depends upon $x_1/x_2$? or how the $x_1$ depends upon $x_2$? Like if I consider the ratio to correspond to $\theta$ and both variables being the function of it? Does this mean I should end my calculation in the last line with $d x_1 (z) / d z$? – Athaeneus Apr 04 '23 at 18:13
  • 1
    @Athaeneus Correct to your first statement; the total derivative accounts for all ways a function can change in response to a variable, but in this case, you have not specified how $x_2$ and $x_1$ relate to each other, so we can't track how $x_1$ changes as $x_1/x_2$ changes. The substitution you mention just relabels things but doesn't solve this problem. Now, if you are treating $x_2$ as independent of $x_1$, i.e. constant in $x_1$, then your total derivative is essentially a partial derivative so $dx_1/d(x_1/x_2)=dx_2(x_1/x_2)/d(x_1/x_2)=x_2d(x_1/x_2)/d(x_1/x_2)=x_2.$ – Golden_Ratio Apr 04 '23 at 18:22

0 Answers0