Domanda del libro di testo
In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.
xy + 2x - 5y = 2, (3, 2)
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Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 3, Problema 3.7.5
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x²(x – y)² = x² – y²
Guida verificata passo dopo passo1
Start by differentiating both sides of the equation with respect to x. The equation is x²(x – y)² = x² – y².
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 = (x - y)².
Differentiate u = x² to get u' = 2x. Differentiate v = (x - y)² using the chain rule: v' = 2(x - y)(1 - dy/dx).
Apply the chain rule to the right side of the equation: differentiate x² to get 2x and differentiate -y² to get -2y(dy/dx).
Set the derivatives of both sides equal to each other and solve for dy/dx. This involves substituting the derivatives found in the previous steps and isolating dy/dx.
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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, often x, while treating other variables, like y, as implicit functions of x. This method is essential when dealing with equations where y cannot be easily isolated.
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Percorso guidato
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, multiplied by the derivative of the inner function. In implicit differentiation, the chain rule is often applied when differentiating terms involving y, as y is considered a function of x.
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05:02Intro to the Chain Rule
Product Rule
The product rule is a technique used to differentiate products of two or more functions. It states that the derivative of a product of two functions u(x) and v(x) is given by u'(x)v(x) + u(x)v'(x). In the given problem, the product rule is necessary to differentiate terms like x²(x – y)², where multiple functions of x are multiplied together.
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05:18The Product Rule
Pratica correlata
Domanda del libro di testo
Slopes on the graph of the tangent function Graph y = tan x and its derivative together on (−π/2, π/2). Does the graph of the tangent function appear to have a smallest slope? A largest slope? Is the slope ever negative? Give reasons for your answers.
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Domanda del libro di testo
The devil’s curve (Gabriel Cramer, 1750) Find the slopes of the devil’s curve y⁴ – 4y² = x⁴ – 9x² at the four indicated points.
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For Exercises 55 and 56, evaluate each limit by first converting each to a derivative at a particular x-value.
lim (x → −1) (x²/⁹ − 1) / (x + 1)
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Domanda del libro di testo
Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?
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Domanda del libro di testo
In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.
x³/² + 2y³/² = 17, (1, 4)
217
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