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The chain rule $\frac{dy}{dx} = \frac{dy}{dt}*\frac{dt}{dx}$ can be regarded as product of slope of $y$ relative to $t$ axis and slope of $x$ relative to $x(t)$ axis. As I read in this question it makes sense to multiply slopes sometimes because if $f(x)=ax+b, g(x)=cx+d$ then the slope of $f(g(x))$ is $ac$. Now I am trying to see if these two ideas relate to each other. If it is so I would like to know what plays roles of $g(x)$ and $f(x)$.

I am asking this to understand chain rule better. Currently, I got that $\frac{dy}{dt}$ part represents the angle between $y'$ and $t$ axis when $\frac{dt}{dx}$ stays for an angle between $x'$ and $x(t)$ axis. Now, I am trying to make sense of multiplication operator between these two terms. The only meaningful explanation of slope product I found in the question I referenced above. In case if that product has nothing to do with function composition I would like to ask about possible meaning of that $*$ operator in the middle of the chain rule.

xevepisis
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    I've always seen it as just multiplication of the two quanities, but I'm excited to see if there is a deeper meaning. – The Count Oct 23 '18 at 21:15

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Great question and well-asked as a new user to the site. Maybe I can give a bit of geometric intuition to help understand why the chain rule has to do with the product of slopes.

So I like to (informally) view differentials $dx$ and $dy$ as representing a small "nudge" in the directions x and y respectively in the xy plane. I think with this in mind, the chain rule makes more sense in terms of multiplication. So $\frac{dy}{dx}$ can be intuitively thought of as "when I nudge x a bit, by a factor of $dx$, by what proportion does this change $y$"? Essentially, this is how you can geometrically think of the derivative: it's the ratio of the nudge of $y$, denoted as $dy$, compared to the nudge of $x$, denoted $dx$.

Okay, so now we have this external parameter, $t$. When we nudge $x$ by some little bit $dx$, $x(t)$ changes by a factor of $\frac{dt}{dx}dx = dt$. But that $dt$ changes $y$ by a factor of $\frac{dy}{dt}dt = dy$. So a nudge in $x$ induces this nudge in $t$ which induces a nudge in $y$. Thus, it makes sense then that the "end effect" is the product of these ratios! For example, say at some point $\frac{dt}{dx} = 3$ and $\frac{dy}{dt} = 2$. This means that $dt = 3 dx$ and $dy = 2dt$. So then $dy=2(3dx)$ and thus $dy=6dx$ and so $\frac{dy}{dx} = 6$, which aligns with what we get with the chain rule.

I find that when starting out in calculus, reasoning about it informally like this with differentials really helps to develop the intuition.

rb612
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  • Thank you for the different perspective. It seems that formula makes sense if you consider derivatives not as slopes but as the rates of change. Then $\frac{dy}{dt}$ is the rate at which $t$ changes when $y$ is changed and $\frac{dt}{dx}$ is the rate at which $x$ changes when $t$ was changed. The product of these changes is final derivative $\frac{dy}{dx}$. Did I understand your point correctly? – xevepisis Oct 23 '18 at 22:50
  • @xevepisis yes, you got it! As a silly but more concrete example, if me taking a step causes my dog to take 3 steps, and for every step my dog takes, my hamster take 2 steps, then every step I take will make my hamster take 6 steps! :) And I highly recommend 3blue1brown where he goes into more depth into this "nudging" intuition. See the video here. – rb612 Oct 23 '18 at 23:12
  • Oh! I forgot that 3blue1brown has videos about calculus. This will help a lot. Thank you! – xevepisis Oct 23 '18 at 23:22