Что такое Правила дифференцирования?

Rules of Differentiation

Sum: (f+g)' = f'+g'. Product: (fg)' = f'g + fg'. Quotient: (f/g)' = (f'g - fg')/g². Chain rule: (f(g(x)))' = f'(g(x))·g'(x)

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📖1 min read📊Level 6🗺️9 subtopics📅April 16, 2026

Author: Бессмертный А.П.

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Power Rule
Derivative of a power function: (x^n)' = n×x^(n-1). Works for any n (integer, fractional, negative). Examples: (x³)' = 3x², (√x)' = 1/(2√x), (1/x)' = -1/x². Basis of differential calculus
Chain Rule
Derivative of a composition of functions: (f∘g)'(x) = f'(g(x))·g'(x). Outer function × derivative of inner. Example: (sin(x²))' = cos(x²)·2x. Repeated application: f(g(h(x))). Basis of backpropagation in neural networks. Leibniz notation: dy/dx = dy/du · du/dx
Quotient Rule
Derivative of a fraction: (f/g)' = (f'·g - f·g')/g² for g ≠ 0. Mnemonic: 'lower derivative of upper minus upper derivative of lower, all over lower squared'. Example: (sin x / x)' = (x·cos x - sin x)/x²
Product Rule
The derivative of a product of two functions: (f·g)' = f'·g + f·g'. Example: (x²·sin x)' = 2x·sin x + x²·cos x. Generalization for n functions is also provided
Implicit Differentiation
Derivative of an implicitly defined function (e.g., x² + y² = 1). Differentiate both sides with respect to x, y treated as a function y(x). Apply chain rule to y. Find dy/dx without explicit expression for y
Quotient Rule
Derivative of a fraction: (u/v)' = (u'v - uv')/v². Mnemonic: 'bottom × derivative of top minus top × derivative of bottom, divided by bottom squared'. Alternative: u/v = u·v⁻¹, apply product rule. Example: (tan x)' = (sin x / cos x)' = 1/cos²x = sec²x
Table of Derivatives
Derivatives of elementary functions. (eˣ)' = eˣ. (aˣ)' = aˣ·ln a. (ln x)' = 1/x. (sin x)' = cos x. (cos x)' = -sin x. (tan x)' = sec²x. (arcsin x)' = 1/√(1-x²). (arctan x)' = 1/(1+x²). Memorization is the basis of differentiation techniques
Chain Rule
Derivative of a composite function: (f(g(x)))' = f'(g(x)) · g'(x). Composition of functions, differentiation "inside out". Example: (sin(x²))' = cos(x²) · 2x. Critical for neural networks (backpropagation)
Logarithmic Differentiation
A method for products/quotients/powers: take ln of both sides, differentiate, and express y'. Simplifies calculations by turning products into sums

Правила дифференцирования — Сумма: (f+g)' = f'+g'.

Произведение: (fg)' = f'g + fg'. Частное: (f/g)' = (f'g - fg')/g².

Предпосылки и причины

Сложная функция: (f(g(x)))' = f'(g(x))·g'(x)..

Ход событий

Развитие определялось действиями участников и обстановкой.

Последствия и значение

Результаты оказали влияние на дальнейшее развитие событий.

❓Часто задаваемые вопросы

Сумма: (f+g)' = f'+g'. Произведение: (fg)' = f'g + fg'. Частное: (f/g)' = (f'g - fg')/g². Сложная функция: (f(g(x)))' = f'(g(x))·g'(x).

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What is Rules of Differentiation?Rules of Differentiation: lecture Rules of Differentiation: overview