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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 3.8.5
3장, 문제 3.8.5

If y = x² and dx/dt = 3, then what is dy/dt when x = –1?

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1
First, identify the given function: y = x². This is the function that relates y and x.
Recognize that you need to find dy/dt, which is the rate of change of y with respect to time t.
Use the chain rule for differentiation, which states that dy/dt = (dy/dx) * (dx/dt).
Differentiate y = x² with respect to x to find dy/dx. The derivative of x² with respect to x is 2x.
Substitute the given values into the chain rule formula: dy/dt = 2x * (dx/dt). Use x = -1 and dx/dt = 3 to find dy/dt.

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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. In this problem, it helps us differentiate y = x² with respect to time t, considering x as a function of t, to find dy/dt.
추천 영상:
가이드 코스
05:14
Finding The Implicit Derivative

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, it allows us to relate dy/dt to dx/dt by differentiating y = x² with respect to t.
추천 영상:
05:02
Intro to the Chain Rule

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to time, given the rate of change of another related quantity. In this scenario, we are given dx/dt and need to find dy/dt when x = -1, using the relationship between x and y provided by the equation y = x².
추천 영상:
가이드 코스
04:16
Intro To Related Rates
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