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I am wondering why we don't learn the multi-variate chain rule in Calculus I? I know the name implies it is more suitable for multi-variable Calculus, but after learning it, I've found it very useful. Notably, one does not need to remember product rule or quotient rule or regular chain rule, and I don't think you would have to learn about logarithmic differentiation either.

So with all these advantages, why don't we teach it?

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    $\begingroup$ How do you get the derivative of $x\sin x$ using that rule? $\endgroup$ – mfl Dec 12 '16 at 14:38
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    $\begingroup$ @mlf You can the derivative of $f(u,v)=uv$, where $u=x$ and $v=\sin x$ $\endgroup$ – Simply Beautiful Art Dec 12 '16 at 14:43
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    $\begingroup$ Even with the chain rule established, logarithmic differentiation gives you a lot for a little. $\endgroup$ – Omnomnomnom Dec 12 '16 at 14:54
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    $\begingroup$ technically this would go on matheducators.SE $\endgroup$ – djechlin Dec 12 '16 at 18:05
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    $\begingroup$ @SimpleArt you seem to be asking why it's not taught in Calc I. you can learn it whenever you want. I think your top answer is good but ME may have answers from people who think about this quite a lot. $\endgroup$ – djechlin Dec 12 '16 at 22:24
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I used to think this, too, until I taught Calculus I.

If you, as a math student and enthusiast, like to see the product rule, etc., as special cases of the multivariate chain rule, then that is good for you and deepens your understanding.

However, my experience has been that reasoning from the general to the specific doesn't always sink in to the novice learner. If the multivariate chain rule is mumbo-jumbo, nothing derived from it is understandable either.

The median student in Calculus I struggles with the concept of function, has trouble working with more than two variables, and can't keep straight whether $\frac{1}{x}$ is the derivative of $\ln x$ or the other way around. I'm not trying to bash Calculus I students; only to recognize that they are in a different place mathematically than we are now, or even than we were when we first learned Calculus I. To reach them, we have to understand where their frontiers are and what is just beyond them.

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    $\begingroup$ Yes, this always true. $\ddot\smile$ probably why I joined this site, to be with more of the same type of math people. $\endgroup$ – Simply Beautiful Art Dec 12 '16 at 14:54
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    $\begingroup$ For some reason I thought for the first quarter of Calculus I the product rule was the definition of the operator to multiply differentials. $\endgroup$ – Joshua Dec 12 '16 at 19:01
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    $\begingroup$ Goodness, this is most popular answer of the day. $\endgroup$ – Simply Beautiful Art Dec 12 '16 at 22:18
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    $\begingroup$ Good job on reaching that! $\endgroup$ – Skeleton Bow Dec 12 '16 at 22:18
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    $\begingroup$ I'd agree for sure but this is one of the many reasons that it doesn't make sense to have mathematics students and physics/engineering students in the same class for the first three (and perhaps most formative) semesters of the undergrad years. The disciplines have two totally different mindsets/goals and usually the class ends up exclusively catering to the physics/engineering crowd. $\endgroup$ – TheLaughingAlgebrist Dec 15 '16 at 19:57

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