I've come across the following expression in my textbook about the chain rule in partial differentiation that I don't quite follow

To be more specific, it's the diferentiation of (6.9) right at the top of page 220

The way I see it, shouldn't it have been

$$\frac{\partial f}{\partial (y+ \eta \alpha)} \frac{\partial (y+ \eta \alpha)}{\partial \alpha} + ... = \frac{\partial f}{\partial (y+ \eta \alpha)} \eta + ...$$

Instead, he differentiated w.r.t. $y$ instead of $(y+ \eta \alpha)$, why is that?

  • 1
    $\begingroup$ The way I understand it, it is using the notation $\delta f/\delta y$ to mean partial derivate wrt the first component, and $\delta f/\delta y'$ to mean partial derivative wrt the second component. $\endgroup$ – Jsevillamol Feb 3 '16 at 11:54
  • $\begingroup$ Could you elaborate further, please? You mean it's using the $\partial$ symbol instead of $\delta$ or that the differentiation is something like $\frac{\partial f}{\partial \alpha} = \frac{\delta f}{\delta y} \frac{\partial (y + \eta \alpha)}{\partial \alpha} + ...$ $\endgroup$ – math84595 Feb 3 '16 at 12:28
  • $\begingroup$ May I ask which textbook you are using? $\endgroup$ – Abstract Analysis Nov 19 at 0:56

If we have a 3-place function $f(y,y',x)$ it is common notation to use $\frac{\partial f}{\partial y}$, $\frac{\partial f}{\partial y'}$ and$\frac{\partial f}{\partial x}$ to mean respectively the first, second and third partial derivatives.

In the example you provide, it is taking the first and second derivatives wrt $\alpha$ (the third does not appear since the third argument of the function does not variate in function of $\alpha$).

Sorry for the typo in the comment, I meant $\partial$ and not $\delta$.

  • $\begingroup$ I understood what you meant, the full differentiation would be $\frac{\partial f (y, y', x)}{\partial \alpha} = \frac{\partial f(y, y', x)}{\partial y}\frac{\partial y}{\partial \alpha} + \frac{\partial f(y, y', x)}{\partial y'}\frac{\partial y'}{\partial \alpha} + \frac{\partial f(y, y', x)}{\partial x}\frac{\partial x}{\partial \alpha}$. However the function in my textbook is $f(y+\alpha \eta, y' + \alpha \eta', x)$, so my whole confusion is why did the author differentiate the components w.r.t. to just $y$ then $y'$ instead of the full terms $y + \alpha \eta$ and $y' + \alpha \eta'$ $\endgroup$ – math84595 Feb 3 '16 at 12:51
  • $\begingroup$ @math84595 It is just notation. The standard would have been to write $\partial f/\partial x$ and $\partial f/\partial y$ for the first and second partial derivatives, but that would make things unclear in this case. $\endgroup$ – Jsevillamol Feb 3 '16 at 12:56
  • $\begingroup$ I see it now. Thanks a lot for your time and help! $\endgroup$ – math84595 Feb 3 '16 at 13:31

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