Where do the power boosts of Kaioken and Super Saiyan come from?

[Doctor.]

This always irked me. We know Kaioken takes the user's Ki and multiplies it and Super Saiyan works through the same principle. But how exactly do they multiply that power? Where's the source of that sudden burst of strength? If you have a cup of water, you can't magically make it overflow.

Transformations in other Shounen are usually explained and even the others in Dragon Ball are explained in some way. Oozaru takes energy from the moon, like filling that cup of water I talked about earlier with more water from a botte. The Elder Kaioshin and Elder Namekian power-ups are outside influences bringing the power you already had, like opening a closed dam by force. Fusion and absorptions take two beings' power and combines them. Freeza's transformations are him just releasing the artificial limiters he placed on his power. So what exactly are Kaioken and Super Saiyan? They certainly don't borrow power from any outside source. Are they just bringing out power from within yourself that you always had but couldn't bring out? If so, why the change in appearance and aura? And plus, while this explanation may be plausible for Super Saiyan, it doesn't really work thematically with Kaioken.

[Polyphase Avatron]

I always thought of Kaio-ken as akin to overclocking a computer - you get increased performance for a limited time at the risk of the whole thing breaking down.

Super Saiyan is more like a complete upgrade. Once it's unlocked your body just becomes much more effective and efficient at generating ki. Something to do with those S-Cells we just learned about recently, most likely.

[Son_Gohan]

The Kaio-Ken draws on dormant power/power they have yet to obtain; Super Saiyan draws from anger/deep-rooted emotion.

[Asura]

Super Saiyan draws from anger/deep-rooted emotion.

Also a reaction to not scratching your back in awhile.

[theherodjl]

Every Saiyan with S Cells has innate potential power that is 400x greater than their ordinary form, the number for SSJ3 which was said in the Daiz to be the power of a Saiyan drawn out to its limits. Kaio-ken on the other hand surges the power of users upwards by however much pressure their bodies can withstand, presumably anyone can learn Kaio-ken so its not something unique to Saiyan potential but general force.
I guess the progression to SSJ3 is like the end result of how strong a muscle can inherently become after hitting its weight limit, while Kaio-ken is pushing a muscle with adrenaline to lift weight beyond its weight limit but at the risk of tearing.

[nickzambuto]

We actually found out the answer to this regarding Super Saiyan recently. Apparently all saiyans have something called S-Cells in very limited supply that are more powerful than regular cells, and the transformation causes them to multiply.

With Kaioken, following your analogy, it would be akin to widening the pipe slightly (bulking body) and increasing the pressure massively, forcing out a larger, wild stream of water. This simultaneously uses up your supply of ki faster, and also strains your body through the ripping muscles and strains your mind and ki through the extremely high pressure forcing ki through too small a vector. Accounting for both the stamina drain and the physical damage. Basically if you have a set supply of ki, it draws more from that well at once than you normally can, so it's dependent on your supply. Super Saiyan however creates brand new power.

[Darkprince410]

The Super Saiyan transformations draw upon the hidden potential of the individual, releasing a value in relation to the user's base form. This is why the Super Saiyan form can't and isn't used in conjunction with Gohan's Ultimate form, as Gohan's Ultimate form draws out his full hidden potential, and thus there's nothing left for the Super Saiyan forms to unlock.

The Kaiou-ken amplifies and increases the individual's existing ki. It doesn't create ki out of nowhere, but it somehow multiplies the individual's existing ki through dangerous methods.

[shadowfox87]

Well, if you go by the law of conservation of energy and mass, energy can't be created or destroyed. Though this is Dragon Ball, energy boosts can create energy out of nowhere. For Super Saiyans, S-cells absorb a concentrated amount of ki from a trigger like rage. This boosts adrenaline / norepinephrine, increases blood flow, etc. to the S-cells. This heavily amplifies "ki" somehow within the user's body. As for Kaioken, it could just be a surge of adrenaline that give a steroid-like effect that increases muscle mass. However, increasing muscle contraction too much can lead to spasms or tetany which is why you would get them in Kaioken.