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If x = y³ – y and dy/dt = 5, then what is dx/dt when y = 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 3.7.10
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
xy = cot(xy)
Guida verificata passo dopo passo1
Start by recognizing that the equation xy = cot(xy) involves both x and y, and y is a function of x. This means we need to use implicit differentiation.
Differentiate both sides of the equation with respect to x. For the left side, use the product rule: d/dx[xy] = x(dy/dx) + y.
For the right side, differentiate cot(xy) with respect to x. Use the chain rule: d/dx[cot(u)] = -csc^2(u) * du/dx, where u = xy. So, differentiate xy with respect to x, which gives: d/dx[xy] = x(dy/dx) + y.
Set the derivatives equal: x(dy/dx) + y = -csc^2(xy) * (x(dy/dx) + y).
Solve for dy/dx by isolating it on one side of the equation. This involves algebraic manipulation to factor out dy/dx and simplify the expression.
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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 as implicit functions of x. This method is essential when dealing with equations like xy = cot(xy), where y is not 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 crucial when differentiating terms involving products or compositions of x and y, such as cot(xy).
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05:02Intro to the Chain Rule
Trigonometric Derivatives
Understanding the derivatives of trigonometric functions is vital for solving problems involving implicit differentiation. For example, the derivative of cot(u) with respect to u is -csc^2(u). When differentiating an equation like xy = cot(xy), knowing these derivatives allows you to correctly apply the chain rule and find dy/dx, especially when trigonometric functions are involved.
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06:35Derivatives of Other Inverse Trigonometric Functions
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