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 44
Find by implicit differentiation.
x² + xy + y² - 5x = 2
Verified step by step guidance1
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 term x² with respect to x, which gives 2x.
For the term xy, apply the product rule: differentiate x to get 1, multiply by y, then differentiate y with respect to x (dy/dx) and multiply by x. This results in y + x(dy/dx).
Differentiate y² with respect to x using the chain rule: 2y(dy/dx).
Differentiate the term -5x with respect to x, which gives -5. Set the derivative of the right side, which is a constant 2, to 0. Combine all these results to form the equation: 2x + y + x(dy/dx) + 2y(dy/dx) - 5 = 0.
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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 variable is not isolated on one side. Instead of solving for y explicitly, 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 needing to express y in terms of x.
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Finding The Implicit Derivative
Chain Rule
The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When applying implicit differentiation, we often encounter terms involving y, which requires us to use the chain rule to differentiate these terms correctly. For example, when differentiating y², we apply the chain rule to get 2y(dy/dx).
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05:02Intro to the Chain Rule
Collecting Like Terms
Collecting like terms is a crucial step in simplifying equations after differentiation. After applying implicit differentiation, we often end up with an equation that contains multiple terms involving dy/dx. By rearranging and combining these terms, we can isolate dy/dx, making it easier to solve for the derivative and understand the relationship between x and y.
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4:46Solving Exponential Equations Using Like Bases
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
Find by implicit differentiation.
x²y² = 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
Slopes and Tangent Lines
b. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope?
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Textbook Question
Airplane takeoff Suppose that the distance an aircraft travels along a runway before takeoff is given by D = (10/9)t², where D is measured in meters from the starting point and t is measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches 200 km/h. How long will it take to become airborne, and what distance will it travel in that time?
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