Textbook Question
Second Derivatives
In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.
2√y = x – y
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 3.8.3
Assume that y = 5x and dx/dt = 2. Find dy/dt
Verified step by step guidance1
Start by identifying the given information: y = 5x and dx/dt = 2. We need to find dy/dt.
Recognize that y is a function of x, and both y and x are functions of time t. This implies we need to use the chain rule for differentiation.
Apply the chain rule: The derivative of y with respect to t, dy/dt, can be found using dy/dt = (dy/dx) * (dx/dt).
Differentiate y = 5x with respect to x to find dy/dx. Since y = 5x, dy/dx = 5.
Substitute dy/dx = 5 and dx/dt = 2 into the chain rule formula: dy/dt = 5 * 2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
A derivative represents the rate at which a function is changing at any given point and is a fundamental tool in calculus. In this context, dy/dt and dx/dt are derivatives that describe how y and x change with respect to time t. Understanding derivatives is crucial for solving problems involving rates of change.
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Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It is essential here because y is a function of x, which is itself a function of t. The chain rule allows us to find dy/dt by multiplying the derivative of y with respect to x (dy/dx) by the derivative of x with respect to t (dx/dt).
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05:02Intro to the Chain Rule
Linear Functions
A linear function is a function of the form y = mx + b, where m and b are constants. In this problem, y = 5x is a linear function with a slope of 5. Understanding linear functions helps in recognizing that the derivative dy/dx is constant, which simplifies the application of the chain rule to find dy/dt.
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07:17Linearization
Related Practice
Textbook Question
In Exercises 41–58, find dy/dt.
y = (1 + cos(2t))⁻⁴
Textbook Question
In Exercises 41–58, find dy/dt.
y = 4 sin(√(1 + √t))
Textbook Question
Derivatives
In Exercises 27–32, find dp/dq.
p = (q sin q) / (q² − 1)
Textbook Question
Find the derivatives of the functions in Exercises 1–42.
𝓻 = ( sin θ )²
( cos θ - 1 )
Textbook Question
Find the first and second derivatives of the functions in Exercises 33–38.
s = (t² + 5t − 1) / t²