#C02 · Calculus & Analysis · Lv.2View in 3D Mind-Map

Derivative Definition & Instantaneous Rate

Isaac Newton & Gottfried Wilhelm Leibniz · 1684 CE

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

\frac{d}{dx}(x^n) = nx^{n-1},\quad \frac{d}{dx}(\sin x) = \cos x

Shrinks the interval h → 0 to turn average secant slope into instantaneous tangent slope, unlocking velocity, acceleration, and function optimization.

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Point x₀

Step h

\frac{f(2+0.5)-f(2)}{0.5} = 4.500 \xrightarrow{h\to 0} f'(2) = 4.000

순간 접선 기울기 f'(2)

4.0000

할선 평균변화율 (Δx=0.5)

4.5000

근사 오차 |평균 - 순간|

0.5000

Prerequisite Path (7 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}

🚀 Next Unlocks (Derived Formulas) (5)

#C03Product Rule & Chain Rule (Backpropagation Core)#C04Fundamental Theorem of Calculus (FTC)#C06Taylor & Maclaurin Series Expansion #C08Partial Derivatives & Gradient Vector (∇f)#C10Differential Equations & Harmonic Oscillator

Geometric & Intuitive Insight

곡선 위의 두 점을 잇는 직선(할선)에서 두 점을 한없이 가깝게 모으면, 곡선에 스치듯 닿는 단 하나의 접선이 되며 그 기울기가 바로 도함수 f'(x)입니다.

Step-by-Step Derivation

Step 1.1단계: f(x) = x^n 일 때 이항정리에 의해 f(x+h) = x^n + n·x^(n-1)h + O(h²) 입니다.
Step 2.2단계: 차분몫 (f(x+h) - f(x)) / h = n·x^(n-1) + O(h) 를 구합니다.
Step 3.3단계: h → 0 극한을 취하면 고차항 O(h)가 사라져 (x^n)' = n·x^(n-1) 이 증명됩니다.

AI & Coding Real-World Use

Physics engine velocity/acceleration finite differences and neural network loss slope

#Numerical Differentiation#Physics Velocity#Optimization Extremum

const diff = (f: (x:number)=>number, x: number, h = 1e-5) => (f(x + h) - f(x - h)) / (2 * h);

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