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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 3 - DerivativesProblema 48
Capitolo 3, Problema 48

Find by implicit differentiation.
x²y² = 1

Guida verificata passo dopo passo
1
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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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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Percorso guidato
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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Intro 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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The Product Rule
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