Build the idea, one step at a time一步步建立理解
Need a math refresher?需要补一点数学基础？
Open only the tool you need; return here to use it in the next step.按需打开一个小工具，再回到本课把它用起来。
Derivatives as local rates导数：局部变化率The chain rule链式法则STEP步骤 1 / 5
Turn a color difference into a scalar把颜色差变成一个数
For one scalar pixel, let prediction be Ĉ and target C*. Squaring their difference makes positive and negative errors both cost something. For RGB MSE, average the three squared errors; for a batch, average across pixels and samples as well. The averaging factors belong in the derivative too.一个标量像素的预测为 Ĉ，目标为 C*。差值平方后，偏高偏低都有代价。RGB MSE 将三个平方误差取平均，批量训练还要对像素与样本取平均。求导时同样不能漏掉这些平均系数。
STEP步骤 2 / 5
Use a tiny renderer you can differentiate by hand先用可以手算求导的微型渲染器
Fix the spatial Gaussian weight at one, use black background and one layer with color c. Its prediction is Ĉ = cα and α = sigmoid(β). Only β is learned here. The sigmoid derivative is α(1 − α): the same raw step has a smaller visible effect near saturation.把空间权重固定为 1，使用黑背景、一个颜色为 c 的层。预测 Ĉ = cα，α = sigmoid(β)，这里只学 β。sigmoid 的导数是 α(1 − α)：接近饱和时，同样原始参数步长引起的可见变化更小。
STEP步骤 3 / 5
Multiply the local conversion rates把局部换算率乘起来
A tiny change in β changes α, which changes Ĉ, which changes L. Multiply these three rates. This is the chain rule. With c = 0.8, β = 0 and target = 0.6, α = 0.5, Ĉ = 0.4 and the gradient is −0.08. Subtracting a negative gradient increases β and therefore opacity.β 微变会引起 α 变化，再改变 Ĉ，最后改变 L。将三段变化率相乘就是链式法则。c = 0.8、β = 0、目标 0.6 时，α = 0.5、Ĉ = 0.4，梯度为 −0.08。减掉负梯度会增大 β，从而增大 opacity。
STEP步骤 4 / 5
Extend from one path to a graph从一条链扩展成计算图
A Gaussian position changes its projected center, its projected covariance, possibly its viewing direction and visibility. All differentiable paths must contribute. In the current direct-RGB evaluator the position gradient includes both projected-center and covariance paths. Overlapping splats also couple gradients through transmittance: a front opacity can reveal or hide a back color.Gaussian 位置会改变投影中心、投影协方差，可能还改变观察方向与可见性。所有可微路径都要贡献梯度。当前直接 RGB 教学器的位置梯度同时包含中心和协方差两条路径。重叠点还通过透射率耦合：前点 opacity 可露出或遮住后点颜色。
STEP步骤 5 / 5
Check derivatives rather than trusting them用数值检查导数
Perturb β by a small positive and negative ε and compare the loss difference. This finite difference should agree with the analytic gradient in a smooth region. Sorting, clipping, alpha thresholds and changing membership create branches; standard splat backward passes differentiate the active computation rather than a smooth global ordering function. Too small ε can also magnify roundoff.给 β 加减一个小 ε，比较损失变化。平滑区域里，有限差分应与解析梯度接近。排序、裁剪、alpha 阈值和成员变化都会产生分支；常见 Splat 反向过程对当前活跃计算求导，而不是对全局排序函数做平滑求导。ε 太小还会放大舍入误差。
PUT NUMBERS IN代入数字
At the initial values, loss is 0.04. A step with η = 1 changes β from 0 to 0.08, α to about 0.51999, prediction to 0.41599 and loss to about 0.03386. No mysterious “AI instruction” is involved—just this chain of arithmetic.在初值下，损失是 0.04。η = 1 的一步把 β 从 0 改为 0.08，α 约为 0.51999，预测约为 0.41599，损失约为 0.03386。这里没有神秘的“AI 指令”，就是这串算术。