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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 3.7.22
3장, 문제 3.7.22

Second Derivatives


In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.


y² – 2x = 1 – 2y

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Start by differentiating both sides of the equation y² - 2x = 1 - 2y with respect to x. Remember to apply implicit differentiation, which means treating y as a function of x.
Differentiate the left side: The derivative of y² with respect to x is 2y(dy/dx) using the chain rule, and the derivative of -2x is -2.
Differentiate the right side: The derivative of 1 is 0, and the derivative of -2y with respect to x is -2(dy/dx) using the chain rule.
Set the derivatives equal: 2y(dy/dx) - 2 = -2(dy/dx). Solve this equation for dy/dx to find the first derivative.
To find the second derivative d²y/dx², differentiate the expression for dy/dx with respect to x again. Use implicit differentiation and substitute dy/dx from the previous step where necessary. Simplify the expression to write d²y/dx² in terms of x and y.

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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. It involves differentiating both sides of an equation with respect to a variable, typically x, while treating other variables as implicit functions of x. This method is essential for equations where y is not isolated.
추천 영상:
가이드 코스
05:14
Finding The Implicit Derivative

First Derivative (dy/dx)

The first derivative, dy/dx, represents the rate of change of y with respect to x. In the context of implicit differentiation, it involves applying the chain rule to differentiate terms involving y, treating y as a function of x. Solving for dy/dx is crucial for understanding the slope of the curve defined by the implicit equation.
추천 영상:
07:09
The First Derivative Test: Finding Local Extrema

Second Derivative (d²y/dx²)

The second derivative, d²y/dx², provides information about the curvature or concavity of the graph of a function. After finding dy/dx using implicit differentiation, d²y/dx² is obtained by differentiating dy/dx again with respect to x, applying the chain rule and product rule as necessary. This derivative helps in analyzing the behavior of the function's graph, such as identifying points of inflection.
추천 영상:
06:02
The Second Derivative Test: Finding Local Extrema
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Second Derivatives


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_____

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