Textbook Question
Assume that y = 5x and dx/dt = 2. Find dy/dt
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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 3.3.34
Find the first and second derivatives of the functions in Exercises 33–38.
s = (t² + 5t − 1) / t²
Verified step by step guidance1
Step 1: Begin by simplifying the function s = (t² + 5t − 1) / t². This can be rewritten as s = t²/t² + 5t/t² − 1/t², which simplifies to s = 1 + 5/t − 1/t².
Step 2: Differentiate the simplified function s = 1 + 5/t − 1/t² with respect to t to find the first derivative. Use the power rule and the derivative of 1/t and 1/t². The derivative of 1 is 0, the derivative of 5/t is -5/t², and the derivative of -1/t² is 2/t³.
Step 3: Combine the derivatives from Step 2 to express the first derivative of s. The first derivative, s', is s' = 0 - 5/t² + 2/t³.
Step 4: Differentiate the first derivative s' = -5/t² + 2/t³ with respect to t to find the second derivative. Again, use the power rule. The derivative of -5/t² is 10/t³, and the derivative of 2/t³ is -6/t⁴.
Step 5: Combine the derivatives from Step 4 to express the second derivative of s. The second derivative, s'', is s'' = 10/t³ - 6/t⁴.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The first derivative indicates the slope of the tangent line to the curve, while the second derivative provides information about the curvature or concavity of the function.
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Quotient Rule
The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If you have a function s = f(t) / g(t), the derivative is given by s' = (f' * g - f * g') / g². This rule is essential when differentiating functions that are expressed as fractions, ensuring accurate results.
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Simplifying Functions
Simplifying functions involves rewriting them in a more manageable form, often to facilitate differentiation or integration. In the context of the given function s = (t² + 5t − 1) / t², simplifying can help identify terms that can be easily differentiated. This process can reveal insights about the function's behavior and make calculations more straightforward.
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Related Practice
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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In Exercises 41–58, find dy/dt.
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Textbook Question
In Exercises 41–58, find dy/dt.
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In Exercises 27–32, find dp/dq.
p = (q sin q) / (q² − 1)
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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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