Textbook Question
Find all points (x, y) on the graph of y = x/(x − 2) with tangent lines perpendicular to the line y = 2x + 3.
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Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 48
Find by implicit differentiation.
x²y² = 1
Verified step by step guidance1
Start by differentiating both sides of the equation with respect to x. The equation is x²y² = 1.
Apply the product rule to the left side of the equation. The product rule states that d(uv)/dx = u'v + uv', where u = x² and v = y².
Differentiate x² with respect to x to get 2x, and differentiate y² with respect to x using the chain rule to get 2y(dy/dx).
Substitute these derivatives back into the product rule: 2x(y²) + x²(2y(dy/dx)) = 0.
Solve for dy/dx by isolating it on one side of the equation. This involves moving terms without dy/dx to the other side and then dividing by the coefficient of dy/dx.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule as necessary. This method is particularly useful for equations like x²y² = 1, where y cannot be easily isolated.
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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, as we often need to differentiate terms involving y, treating y as a function of x.
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05:02Intro to the Chain Rule
Product Rule
The product rule is a formula used to differentiate products of two functions. It states that if you have two functions u(x) and v(x), the derivative of their product is given by u'v + uv'. In the context of the equation x²y² = 1, applying the product rule is necessary when differentiating terms like x² and y², as they are multiplied together, requiring careful application of this rule to obtain the correct derivative.
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05:18The Product Rule
Related Practice
Textbook Question
Slopes and Tangent Lines
a. Horizontal tangent lines Find equations for the horizontal tangent lines to the curve y = x³ − 3x − 2. Also find equations for the lines that are perpendicular to these tangent lines at the points of tangency.
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Textbook Question
In Exercises 51 and 52, find dp/dq.
q = (5p² + 2p)⁻³/²
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Textbook Question
Assume that functions f and g are differentiable with f(1) = 2, f'(1) = −3, g(1) = 4, and g'(1) = −2. Find the equation of the line tangent to the graph of F(x) = f(x)g(x) at x = 1.
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Textbook Question
Quadratics having a common tangent line The curves y = x² + ax + b and y = cx − x² have a common tangent line at the point (1,0). Find a, b, and c.
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Textbook Question
Find by implicit differentiation.
x² + xy + y² - 5x = 2
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