Question

could someone explain to me the chain rule in calculus the most simple way possible... I've had it explained so many times, but i still can't seem to get it..

Answer 1

the chain rule is simply allowing for all the stuff that happens when you start topling a chain of dominoes

Answer 2

peel it apart like an onion and work each section by itself, and then combine it all together again

Answer 3

tan(\(\sqrt{3x^2}\)) for example: it has 3 parts to it

tan(...) ; \(\sqrt{(...)}\); and 3x^2

Answer 4

tan(..) derives to sec^2(....); sec^2(\(\sqrt{3x^2}\))

\(\sqrt{(...)}\) derives to \(\frac{1}{2\sqrt{(...)}}\) ; \(\frac{1}{2\sqrt{(3x^2)}}\)

\(3x^2\) derives to \(6x\); 6x

now multiply all the derivatives together to get:
\[ \frac{6x\ sec^2(\sqrt{3x^2})}{2\sqrt{(3x^2)}}\]

Answer 5

the simplest chain rule you already do; for example: \(x^7\)

this is really;

\((....)^7\) derives to ; \(7(....)^6\); \(7(x)^6\)

and x; which derives to; 1

\[7x^6 * 1 = 7x^6\]

Answer 6

change this to perhaps:
\[(x^2)^6\]

this is equal to x^12 which derives to 12x^11 right?

the chain rule would be:

6(x^2)^5 * 2x

6x^10 * 2x = 12x^11

Answer 7

happy with that blair? maybe i can help a bit more

Answer 8

that is awesome but anymore explanations would be greatly appreciated!

Answer 9

when i think of the chain rule; I imagine a box that has a few gears that are intermeshed; by turning one gear, ALL of the others turn as well. The final outcome is determined by each gears part so that a small turn to begin with can have huge consequences at the end.

The chain rule (gear rule?) is just combining all the effects that each seperate gear plays along the way ...

Answer 10

the 'gear' rule then for just one gear is what you normally consider as a derivative, since there is only one function to keep track of :)

2 gears would have to consider each ones effect...

the more gears you have, the more complicated it becomes, but as long as you focus on each one seperately, and then combine all the efforts, you get a total result

Answer 11

thank you so much! That really helps to think of it like that!