Domanda del libro di testo
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x³ + y³ = 18xy
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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 3.6.3
Derivative Calculations
In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).
y = sin u, u = 3x + 1
Guida verificata passo dopo passo1
First, identify the functions involved: y = sin(u) and u = 3x + 1. We need to find dy/dx using the chain rule.
The chain rule states that dy/dx = dy/du * du/dx. So, we need to find the derivatives dy/du and du/dx separately.
Calculate dy/du: Since y = sin(u), the derivative of y with respect to u is dy/du = cos(u).
Calculate du/dx: Since u = 3x + 1, the derivative of u with respect to x is du/dx = 3.
Combine the derivatives using the chain rule: dy/dx = dy/du * du/dx = cos(u) * 3. Substitute u = 3x + 1 into the expression to get dy/dx = cos(3x + 1) * 3.
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Chain Rule
The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that if you have a function y = f(u) and u = g(x), then the derivative dy/dx is found by multiplying the derivative of f with respect to u, f'(u), by the derivative of u with respect to x, g'(x). This allows us to differentiate complex functions by breaking them down into simpler parts.
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Intro to the Chain Rule
Derivative of Sine Function
The derivative of the sine function, sin(u), with respect to u is cos(u). This is a basic derivative rule in calculus, which is essential for solving problems involving trigonometric functions. When applying the chain rule, knowing the derivative of sin(u) helps in finding the derivative of composite functions involving sine.
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03:53Derivatives of Sine & Cosine
Linear Function Derivative
The derivative of a linear function u = 3x + 1 with respect to x is simply the coefficient of x, which is 3. Linear functions are straightforward to differentiate, and understanding this concept is crucial when applying the chain rule to find dy/dx for composite functions where one part is linear.
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Percorso guidato
07:17Linearization
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