Base balance
Base damage of a Daedric Sword is 14. You get 100% increase in damage from 5/5 Armsman and 50% increase in damage from 100 One-handed. 14 * 2.0 * 1.5 = 42 damage. Swords swing once every second. That's 42 DPS.
Base damage of a Daedric Bow is 19, damage of a Daedric Arrow is 24. Again, you get 100% increase in damage from 5/5 Marksman and another 50% from 100 Archery skill. (19 + 24) * 2.0 * 1.5 = 129. Firing speed of a bow is always at least one second(nock arrow 0.4 second, recoil 0.6 second) plus extra time to pull the string all the way back which depends on bow's weight. Complete formula for firing time is: 1 + [ 0.4 * (1 + 0.1 * bow weight) / Quick shot perk ] seconds. Quick shot perk reduces draw time by 35% so that's 1.35. For Daedric Bow with weight value of 18 we get final firing time of: 1 + [ 0.4 * (1+ 1.8) / 1.35 ] ~= 1.83. So our bow fires once every 1.83 seconds, which gives us DPS value of ~70.48.
Finally we have Destruction spells. Incinerate does 60 damage. The damage can be increased with 2/2 Augmented Flames by 50%. 60 * 1.5 = 90. You cast almost every spell(except for stream spells like Flames) once every second. That's 90 DPS. You can, of course, cast it with both hands to double that value to 180 DPS.
Melee: 42 DPS.
Archery: ~70.48DPS.
Destruction: 90/180 DPS.
So Destruction does most damage per second. By far(very far).
Legendary balance
Let's Smith our weapons to Legendary. Legendary Daedric Sword base damage is 24. 24 * 2.0 * 1.5 = 72 DPS. Legendary Daedric Bow does 29 damage, so the damage formula becomes (29 + 24) * 2.0 * 1.5 = 159. Attack speed doesn't change, so we get DPS of 159 / 1.83 = 86.88.
Legendary Melee: 72 DPS.
Legendary Archery: ~86.88.
Destruction: 90/180 DPS.
That's right - even if you max another non-combat skill and upgrade your weapons to Legendary, Destruction will still deal most DPS without even resorting to casting it with both hands.
OMG Destruction is imba
So now instead of "Why is Destruction so weak" you might ask, "Why is Destruction so strong"?
Simple. The combat skills are balanced not only by how much DPS they do, but also by many other factors.
Melee: Does least DPS and does it only at very close range, but doesn't require any resource to do it. Press attack button, receive DPS, always. Also has power attacks to slightly increase your DPS for a few moments.
Archery: Does more DPS than melee and does it at range. Hovewer archery relies heavily on quality arrows to do damage, and the difference between arrows in damage is huge. Iron Arrow does 8 damage, Daedric Arrow does 24 damage - 3x as much. In comparison, Iron Sword does 7 damage and Daedric Sword does 14, which is just 2x as much. So to deliver good DPS you need to use good arrows, which aren't very common, especially Daedric ones; you will almost certainly be forced to buy them, which means you spend a resource(money) to keep doing high DPS. That's a way to balance higher damage output.
Destruction: Over twice as much DPS as Archery and it's ranged, but again, you're spending a resource. Now theoretically the resource doesn't cost you anything, as Magicka regenerates completely for free, but as anyone knows the regen rate is much slower during combat and as any mage will confirm, running out of Magicka during combat is a Bad Thing. Balanced? Yes; it's a sort of high risk - high reward combat style.
Other things to disprove:
High level Destruction spells are far too costly to be used constantly.
That's why there are perks to reduce casting costs and the skill level itself reduces it further, and mage gear reduces it yet again and incresaes your Magicka pool/regeneration which helps again. From perks and 100 Destruction skill alone, the casting cost of Incinerate will be reduced from 171 to 51.3 Magicka. For a mage with 600 Magicka that's over 12 casts(your Magicka does regenerate even during combat; with 600 Magicka it's 6/second during combat, which means you get enough Magicka to cast another Incinerate in 8.55 seconds). And again, that is all without any mage-specific gear.
But I don't have any stuff that increases my Magicka or reduces the cost. And I spent all my points on Health and Stamina.
Do you also expect to be a good warrior or archer without a weapon or armor and Health/Stamina? This is stupid.
You get 300% damage from skill/perks in melee/archery and only 50% damage increase from Destruction skill/perks so it svcks.
Irrelevant, because the BASE DAMAGE of Destruction spells is much higher than base damage of weapons, so it doesn't need to be increased as much with skills/perks. Look at the numbers again, a Legendary Daedric Sword does 24 base damage, Incinerate - 60 base damage. That's 250% damage of the sword. You might just as well argue that Incinerate is overpowered because it does 2.5x times the damage of a sword, completely ignoring the fact that damage of the sword can be tripled with skills and perks; see just how pointless that is? Destruction works differently from martial skills, deal with it.
You say that you need to buy arrows which is a balancing factor for Archery, but since there's no problem with money in Skyrim that's not really a problem.
But that argument works just as well for Melee and Destruction. If money is not a factor, buy yourself Stamina potions to spam power attacks over and over or Magicka potions to cast Destruction spells forever.
But you can use power attacks to increase your DPS/I can stagger enemies with Power Shot/my every attack in the game ever is a sneak attack(lol).
I also ignored Impact. 'nuff said.
You're ignoring the fact that you can Enchant th-
Yes. That's because Enchanging is a broken skill that is in a dire need of a severe nerf. Or if you're one of those roleplaying people who can't comprehend balance, or giving the player viable options, in an RPG, which is all about giving the player viable options, it's a skill that should never be used. You could print this page and take a dump on it and that would be a quite accurate metaphor to what Enchanting currently does for the balance in the game.
