#C08 · Calculus & Analysis · Lv.3View in 3D Mind-Map

Partial Derivatives & Gradient Vector (∇f)

Euler, Hamilton & Lagrange · 1846 CE

\nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \dots, \frac{\partial f}{\partial x_n}\right)

D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u},\quad \Delta f = \nabla^2 f

Assembles partial derivatives across all variables into the gradient vector ∇f, pointing in the exact direction of steepest ascent on a multidimensional surface.

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Coord x

Coord y

f(x,y)=x^2+2y^2 \implies \nabla f(2,1.5) = (4.0,\; 6.0)

그래디언트 ∇f = (2x, 4y)

(4.0, 6.0)

최대 경사 크기 ||∇f||

7.211

경사하강 방향 -∇f

(-4.0, -6.0)

Prerequisite Path (12 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#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

STEP 5#L01Vector Operations, L1/L2 Norms & Normalization

\|\mathbf{v}\|_2 = \sqrt{\sum_{i=1}^{n} v_i^2}

STEP 6#A06Logarithm Laws & Change of Base

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

STEP 7#A08Binomial Theorem & Pascal's Triangle

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

STEP 8#C01Limits, Continuity & Euler's Number e

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

STEP 9#G022D/3D Euclidean Distance Formula

d(P

STEP 10#C02Derivative Definition & Instantaneous Rate

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

STEP 11#C03Product Rule & Chain Rule (Backpropagation Core)

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

STEP 12#C08Partial Derivatives & Gradient Vector (∇f)

\nabla f = \left(\frac{\partial f}{\partial x_1}

🚀 Next Unlocks (Derived Formulas) (2)

#P08Pearson Correlation & Linear Regression (OLS)#P10Softmax Function & Stochastic Gradient Descent

Geometric & Intuitive Insight

안개 낀 산에서 가장 빨리 정상으로 올라가려면 발밑의 x축 경사(∂f/∂x)와 y축 경사(∂f/∂y)를 합친 벡터 ∇f 방향으로 걸으면 되고, 가장 빨리 골짜기(오차 최소점)로 내려가려면 반대 방향 -∇f 로 걸으면 됩니다!

Step-by-Step Derivation

Step 1.1단계: 다변수 함수 f(x, y)에서 y를 상수로 고정하고 x에 대해서만 미분한 값을 편미분 ∂f/∂x 라 합니다.
Step 2.2단계: 단위벡터 u 방향으로의 방향도함수는 연쇄법칙에 의해 D_u f = (∂f/∂x)u_x + (∂f/∂y)u_y = ∇f · u 입니다.
Step 3.3단계: 내적 공식 ∇f · u = ‖∇f‖ cos θ 에 의해 θ = 0° (즉 u가 ∇f와 같은 방향)일 때 변화율이 최대가 됩니다.

AI & Coding Real-World Use

LLM/Deep Learning loss optimization (-∇L) and computer vision Sobel edge detection gradients

#Gradient Descent -∇L#Computer Vision Sobel#Physics Field ∇

const grad2D = (a:number, b:number, x:number, y:number) => ({ gx: 2*a*x, gy: 2*b*y });

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