#C03 · Calculus & Analysis · Lv.3View in 3D Mind-Map

Product Rule & Chain Rule (Backpropagation Core)

Gottfried Wilhelm Leibniz · 1676 CE

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx},\quad (uv)' = u'v + uv'

(f(g(x)))' = f'(g(x)) \cdot g'(x)

Decomposes the derivative of nested composite functions into a product of local derivatives—the exact mathematical engine of Backpropagation in deep neural networks.

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Inner slope a

Outer power n

Input x

\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx} = 3(3.0)^{2} \cdot (2) = 54.00

합성미분 dy/dx (역전파)

54.000

외부 미분 dy/du = 3u^2

27.000

내부 미분 du/dx = a

2.000

Prerequisite Path (8 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 2#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 3#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 4#A06Logarithm Laws & Change of Base

\log_a(xy) = \log_a x + \log_a y

STEP 5#A08Binomial Theorem & Pascal's Triangle

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

STEP 6#C01Limits, Continuity & Euler's Number e

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828

STEP 7#C02Derivative Definition & Instantaneous Rate

f'(x) = \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

STEP 8#C03Product Rule & Chain Rule (Backpropagation Core)

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

🚀 Next Unlocks (Derived Formulas) (3)

#C05Integration by Substitution & by Parts #C08Partial Derivatives & Gradient Vector (∇f)#P10Softmax Function & Stochastic Gradient Descent

Geometric & Intuitive Insight

톱니바퀴 A가 1바퀴 돌 때 B가 3바퀴 돌고(du/dx=3), B가 1바퀴 돌 때 C가 2바퀴 돈다면(dy/du=2), A가 1바퀴 돌 때 C는 2 × 3 = 6바퀴 돕니다(dy/dx = dy/du · du/dx).

Step-by-Step Derivation

Step 1.1단계: u = g(x), y = f(u) 일 때 증분 비율은 Δy/Δx = (Δy/Δu) · (Δu/Δx) 로 분리됩니다.
Step 2.2단계: Δx → 0 일 때 연속성에 의해 Δu → 0 이므로 각각의 극한이 도함수 f'(u) 와 g'(x) 로 수렴합니다.
Step 3.3단계: 다층 신경망 y = f_L(...f_2(f_1(x))) 에서도 출력 오차부터 입력층까지 국소 미분을 곱해 역전파합니다.

AI & Coding Real-World Use

Deep neural network Backpropagation and PyTorch/TensorFlow Autograd computational graphs

#Deep Learning Backprop#PyTorch Autograd#Neural Net Training

const dL_dw = dL_dy * dy_du * du_dw; // Backpropagation Chain Rule

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