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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 3.7.1
3장, 문제 3.7.1

Differentiating Implicitly


Use implicit differentiation to find dy/dx in Exercises 1–14.


x²y + xy² = 6

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1
Start by differentiating both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating terms involving y, use implicit differentiation.
Differentiate the first term x²y. Apply the product rule: d/dx(x²y) = x²(dy/dx) + y(2x).
Differentiate the second term xy². Again, use the product rule: d/dx(xy²) = x(2y(dy/dx)) + y².
Differentiate the right side of the equation, which is a constant: d/dx(6) = 0.
Combine all the differentiated terms: x²(dy/dx) + y(2x) + x(2y(dy/dx)) + y² = 0. Solve for dy/dx by isolating it on one side of the equation.

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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 differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for y in terms of x, we differentiate both sides of the equation with respect to x, treating y as a function of x. This allows us to find dy/dx without isolating y, which is particularly useful for complex relationships.
추천 영상:
05:14
Finding The Implicit Derivative

Product Rule

The product rule is a fundamental differentiation rule used when differentiating the product of two functions. It states that if u and v are functions of x, then the derivative of their product is given by d(uv)/dx = u'v + uv'. In the context of implicit differentiation, this rule is essential when differentiating terms that involve products of x and y, ensuring that both functions are accounted for.
추천 영상:
05:18
The Product Rule

Chain Rule

The chain rule is a key principle in calculus that allows us to differentiate composite functions. It states that if a function y is dependent on u, which in turn is dependent on x, then dy/dx = (dy/du) * (du/dx). In implicit differentiation, the chain rule is applied when differentiating terms involving y, as we must multiply by dy/dx to account for the dependence of y on x.
추천 영상:
05:02
Intro to the Chain Rule
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