I could write Solidity, sure—but protocol-level knowledge? The kind that separates a developer from a security researcher? That’s where I was missing depth.

That’s why I’m writing this piece.

Uniswap V2 is one of the most integrated DeFi protocols, and understanding its mechanics isn’t optional. Whether you’re building on it or auditing a project that relies on it, you need a solid grasp of how the constant product formula works.

Why Uniswap V2 Still Matters

Uniswap V2 might feel “old” compared to the newer iterations of AMMs, but it’s far from irrelevant.

It still powers countless protocols, serves as a building block for on-chain liquidity, and remains one of the simplest, clearest examples of automated market making in action.

For auditors, this simplicity is deceptive. Many integrations still misapply or misunderstand how V2 handles pricing, swaps, and reserves—and those gaps often lead to exploits.

For developers, knowing V2 at a formulaic level means you can reason about liquidity, slippage, and execution risk without guesswork.

The Constant Product Formula

At the core of Uniswap V2 is a deceptively simple equation:

x * y = k

Here:

• x = reserve of token X in the pool

• y = reserve of token Y in the pool

• k = constant (the product of reserves)

Whenever someone swaps one token for the other, the reserves change, but the product must remain constant. That’s the invariant keeping the pool balanced.

Let’s walk through what happens when a trader adds Δx of token X to the pool and wants to receive Δy of token Y in return.

Reserves Before and After a Swap

Before the trade: the reserves are (x, y)

After the trade: the reserves become (x + Δx, y – Δy).

Here:

• Δx represents the amount of token X being added to the pool, and

• Δy represents the amount of token Y that is being removed from the pool.

Enforcing the Invariant

Remember, the product must stay constant, hence, we calculate the product of this new change after the swap to ensure it matches x * y which is the value that must remain constant.

(x + Δx) * (y – Δy) = x * y

Solving for Δy

Expanding and simplifying gives:

y – Δy = (x * y) / (x + Δx)

So:

Δy = y – (x * y) / (x + Δx)

This is the amount of token Y the trader receives without fees.

Accounting for the 0.3% Fee

In Uniswap V2, a 0.3% fee is charged on the input amount.

This means that only Δx * 0.997 actually contributes to the swap.

Taking this to mind, the formula becomes:

Δy = y – (x * y) / (x + (Δx * 0.997))

The Invariant in Practice

Now, it is important to understand that if there were no fees, the swap would keep the product strictly constant:

(x + Δx) * (y – Δy) = x * y

But Uniswap V2 charges a 0.3% fee on the input as we have already discussed. That means not all of Δx is used for balancing reserves—only Δx * 0.997 contributes.

As a result, after the swap:

x_new * y_new > x_old * y_old

This means that the product doesn’t just stay constant. It increases slightly. That extra value is what counts as liquidity provider (LP) fees.

This is why the Uniswap V2 core contract enforces the invariant as:

require(x_new * y_new >= x_old * y_old, "UniswapV2: K");

The >= ensures that:

• Swaps never reduce k (which would break pool integrity), and

• The system naturally captures fees by letting k grow over time.

This is why it’s completely appropriate to rewrite the formula as:

x * y >= k

Instead of the original:

x * y = k

It is a more accurate reflection of how the invariant behaves in practice.

Example: Swapping ETH for USDC

Suppose we have a WETH/USDC pool with the following reserves:

• x = 100 WETH

• y = 200,000 USDC

This means that the pool price is currently 1 WETH = 2,000 USDC.

Now, a trader wants to swap Δx = 10 WETH into the pool to receive USDC.

1. Apply the fee

   Only Δx * 0.997 = 9.97 WETH counts toward the swap.

2. Plug into the formula

    Δy = y – (x * y) / (x + (Δx * 0.997))
    Δy = 200,000 - (100 * 200,000) / (100 + 9.97)
    Δy = 200,000 - (20,000,000 / 109.97)
    Δy = 200,000 - 181,868
    Δy = 18,132
   

3. Interpretation

   The trader receives about 18,132 USDC.

   Notice it’s not the full 10 * 2000 = 20,000 USDC because of two factors:

   • The price impact: pushing reserves out of balance worsens the rate, and

   • The fee: 0.3% cut built into V2.

After the swap, the new reserves are:

• WETH: 100 + 10 = 110

• USDC: 200,000 – 18,132 ≈ 181,868

And the new implied price is closer to 1 WETH ≈ 1,653 USDC.

This is how Uniswap V2 enforces slippage: the bigger your trade relative to the pool, the worse your execution.

Also, if you calculate the product of the reserves now, the formula will start making sense.

• k_old = 100 × 200,000 = 20,000,000

• k_new = 110 × 181,868 ≈ 20,005,480 > k_old

Why This Matters

The constant product formula is the foundation for how swaps, slippage, and liquidity provisioning work on-chain in Uniswap V2.

For auditors, that means being able to spot when a project integrates Uniswap V2 incorrectly—for example, forgetting to factor in the fee or miscalculating Δy.

For developers, understanding the formula means you can reason about trade execution, slippage, and pool health without relying on external libraries or guesswork.

Bottom line, once you understand it, the attack surfaces and integration risks become obvious.

Conclusion

Uniswap V2 may look simple on the surface, but its constant product formula drives some of the most important dynamics in DeFi. From the way trades settle, to how liquidity moves, to how attackers manipulate integrations. It all comes back to x * y = k or rather, x * y >= k

If you’re building, you now have the baseline for reasoning about swaps, slippage, and pricing.

If you’re auditing, you now know exactly where to start looking for risks.

This article isn’t meant to be the final word on AMMs. It’s meant to be a reference point you can always come back to.

And yes, I know there's a lot of these formulas/calculations flying around here, and this may be confusing to some of you at first glance, but do not worry. I too found it slightly difficult at first — mostly because I hate maths, but trust me, go over it again, take a pen and try writing out the formula on your own and solving for Δy. You'll see that in the end, it's pretty basic stuff.