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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 3 - Derivatives문제 3.8.8
3장, 문제 3.8.8

5–8. Calculate dy/dx using implicit differentiation.
e^y-e^x = C, where C is constant

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1
Start by differentiating both sides of the equation with respect to x. The equation is e^y - e^x = C. Since C is a constant, its derivative is 0.
Differentiate e^y with respect to x. Use the chain rule: the derivative of e^y with respect to y is e^y, and then multiply by dy/dx because y is a function of x.
Differentiate e^x with respect to x. The derivative of e^x with respect to x is simply e^x.
Set up the equation from the derivatives: e^y * (dy/dx) - e^x = 0.
Solve for dy/dx by isolating it on one side of the equation: e^y * (dy/dx) = e^x, then divide both sides by e^y to get dy/dx = e^x / e^y.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념

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

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. Instead of isolating y, we differentiate both sides of the equation with respect to x, applying the chain rule when differentiating terms involving y. This method is particularly useful for equations where y cannot be easily expressed as a function of x.
추천 영상:
가이드 코스
05:14
Finding The Implicit Derivative

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is essential in implicit differentiation when dealing with terms involving y.
추천 영상:
05:02
Intro to the Chain Rule

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = e^x, where e is the base of the natural logarithm. These functions have unique properties, such as their derivative being equal to the function itself. In the context of the given equation, understanding how to differentiate e^y and e^x is crucial for applying implicit differentiation correctly and finding dy/dx.
추천 영상:
6:13
Exponential Functions
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