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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 3 - Derivatives문제 3.6.3
3장, 문제 3.6.3

Derivative Calculations


In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).


y = sin u, u = 3x + 1

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First, identify the functions involved: y = sin(u) and u = 3x + 1. We need to find dy/dx using the chain rule.
The chain rule states that dy/dx = dy/du * du/dx. So, we need to find the derivatives dy/du and du/dx separately.
Calculate dy/du: Since y = sin(u), the derivative of y with respect to u is dy/du = cos(u).
Calculate du/dx: Since u = 3x + 1, the derivative of u with respect to x is du/dx = 3.
Combine the derivatives using the chain rule: dy/dx = dy/du * du/dx = cos(u) * 3. Substitute u = 3x + 1 into the expression to get dy/dx = cos(3x + 1) * 3.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that if you have a function y = f(u) and u = g(x), then the derivative dy/dx is found by multiplying the derivative of f with respect to u, f'(u), by the derivative of u with respect to x, g'(x). This allows us to differentiate complex functions by breaking them down into simpler parts.
추천 영상:
05:02
Intro to the Chain Rule

Derivative of Sine Function

The derivative of the sine function, sin(u), with respect to u is cos(u). This is a basic derivative rule in calculus, which is essential for solving problems involving trigonometric functions. When applying the chain rule, knowing the derivative of sin(u) helps in finding the derivative of composite functions involving sine.
추천 영상:
03:53
Derivatives of Sine & Cosine

Linear Function Derivative

The derivative of a linear function u = 3x + 1 with respect to x is simply the coefficient of x, which is 3. Linear functions are straightforward to differentiate, and understanding this concept is crucial when applying the chain rule to find dy/dx for composite functions where one part is linear.
추천 영상:
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07:17
Linearization
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