Why do we multiply by the derivative when taking the derivative?

Question

For example:

$$y=3x^2$$ $$dy=6xdx$$

To me, it makes sense to just read it as "the derivative of y is $6x$". I don't fully understand the concept of the derivative, but it just seems strange that we multiply by $dx$. Why not add $dx$? Why not subtract/divide by it, etc...

From Wikipedia:

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable).

The notation is supposed to reflect the fact that it is the limit of a difference quotient. — Edward Evans, Sep 24 '16 at 18:23
Actually, the derivative of $y$ with respect to $x$ is not $\mathrm dy$, but rather $\frac{\mathrm dy}{\mathrm dx}$ — Hagen von Eitzen, Sep 24 '16 at 18:23
Related: http://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio — GoodDeeds, Sep 24 '16 at 18:26
Are you asking this because of u-substitution? And I think my teacher said it had something to do with differential forms: https://en.wikipedia.org/wiki/Differential_form — Simply Beautiful Art, Sep 24 '16 at 18:31

score 1 · Accepted Answer · answered Sep 24 '16 at 18:38

1

Note that this is where we get it from:

$$y=3x^2$$

$$\require{cancel}\frac{dy}{dx}=6x\\\frac{dy}{\cancel{dx}}\cancel{\times dx}=6x\times dx$$

$$\color{green}{dy}=\color{red}{6x\ dx}$$

This form is most suitable for $u$-substitution, for example, since derivatives and integrals are 'inverses' of each other. This way, you can change the variable you are integrating with respect to:

$$\int\color{red}{6x\ dx}=\int\color{green}{dy}$$

answered Sep 24 '16 at 18:38

Simply Beautiful Art

74,685

This answer plus the linked answer by @GoodDeeds really helped me understand. Thank you both! – whatwhatwhat Sep 24 '16 at 18:40
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@whatwhatwhat :) cheers! – Simply Beautiful Art Sep 24 '16 at 18:41
That is an interesting cancellation you made. A similar compuation yields $$\frac{19}{95} = \frac{1\cancel{9}}{\cancel{9}5} = \frac 15.$$ – Umberto P. Sep 24 '16 at 20:02
@UmbertoP. :3 a very famous cancellation indeed. I like it. – Simply Beautiful Art Sep 24 '16 at 20:24
But more seriously, it is loosely because derivatives and integrals are 'inverses' in a sense, and so the cancellation proceeds like that. – Simply Beautiful Art Sep 24 '16 at 20:25

Why do we multiply by the derivative when taking the derivative?

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