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How is lim x🠂0 sin x/x used in this section?
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.8.35
Use implicit differentiation to find dy/dx.
x3 = (x + y) / (x - y)
Guida verificata passo dopo passo1
Start by differentiating both sides of the equation with respect to x. The equation is x^3 = (x + y) / (x - y).
Apply the power rule to differentiate x^3 with respect to x, which gives 3x^2.
For the right side, use the quotient rule for differentiation. If u = x + y and v = x - y, then the derivative of u/v is (v * du/dx - u * dv/dx) / v^2.
Differentiate u = x + y with respect to x, which gives du/dx = 1 + dy/dx. Differentiate v = x - y with respect to x, which gives dv/dx = 1 - dy/dx.
Substitute du/dx and dv/dx into the quotient rule formula and set the derivatives equal: 3x^2 = ((x - y)(1 + dy/dx) - (x + y)(1 - dy/dx)) / (x - y)^2. Solve for 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 y in terms of x, 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 isolating y, which is particularly useful for complex relationships.
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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. When using implicit differentiation, we apply the chain rule to account for the derivative of y with respect to x, denoted as dy/dx. This means that when differentiating terms involving y, we multiply by dy/dx to reflect the dependency of y on x.
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05:02Intro to the Chain Rule
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to isolate variables or terms. In the context of implicit differentiation, after differentiating both sides of the equation, we often need to manipulate the resulting expression to solve for dy/dx. This may include combining like terms, factoring, or moving terms across the equation to achieve the desired form.
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05:25Determine Continuity Algebraically
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