Textbook Question
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x³ + y³ = 18xy
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 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
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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07:17Linearization
Related Practice
Textbook Question
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x²y + xy² = 6
Textbook Question
Find the derivatives of the functions in Exercises 19–40.
q = sin(t / (√t + 1))
Textbook Question
Differentiating Implicitly
Use implicit differentiation to find dy/dx in Exercises 1–14.
x = sec y
Textbook Question
In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.
h(t) = t³ + 3t, (1, 4)
Textbook Question
Find the points on the curve y = tan x, -π/2 < x < π/2, where the normal line is parallel to the line y = -x/2. Sketch the curve and normal lines together, labeling each with its equation.