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
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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.
Recommended video:
05:44
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).
Recommended video:
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.
Recommended video:
07:17Linearization
Related Practice
Textbook Question
In Exercises 41–58, find dy/dt.
y = (1 + cos(2t))⁻⁴
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Textbook Question
In Exercises 41–58, find dy/dt.
y = 4 sin(√(1 + √t))
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Textbook Question
Tangent Lines
In Exercises 35–38, graph the curves over the given intervals, together with their tangent lines at the given values of x. Label each curve and tangent line with its equation.
y = sin x, −3π/2 ≤ x ≤ 2π
x = −π, 0, 3π/2
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Textbook Question
Find the first and second derivatives of the functions in Exercises 33–38.
s = (t² + 5t − 1) / t²
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Textbook Question
In Exercises 43–50, find by implicit differentiation.
y² = x .
x + 1
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