Textbook Question
Assume that y = 5x and dx/dt = 2. Find dy/dt
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 3.6.54
In Exercises 41–58, find dy/dt.
y = 4 sin(√(1 + √t))
Verified step by step guidance1
First, identify the function y = 4 sin(√(1 + √t)). We need to find dy/dt, which involves differentiating y with respect to t.
Notice that y is a composite function. It involves the sine function, a square root, and another square root inside. We will use the chain rule to differentiate it.
Start by differentiating the outer function: y = 4 sin(u), where u = √(1 + √t). The derivative of sin(u) with respect to u is cos(u). Therefore, dy/du = 4 cos(u).
Next, differentiate u = √(1 + √t) with respect to t. This requires using the chain rule again. Let v = 1 + √t, so u = √v. The derivative of √v with respect to v is 1/(2√v).
Finally, differentiate v = 1 + √t with respect to t. The derivative of √t with respect to t is 1/(2√t). Combine all these derivatives using the chain rule to find dy/dt.
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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 technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative dy/dx is f'(g(x)) * g'(x). In this problem, the chain rule helps differentiate the nested functions within y = 4 sin(√(1 + √t)).
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Intro to the Chain Rule
Derivative of Sine Function
The derivative of the sine function is crucial for solving this problem. The derivative of sin(u) with respect to u is cos(u). When differentiating y = 4 sin(√(1 + √t)), this rule is applied to find the derivative of the sine component, which is part of the composite function.
Recommended video:
03:53Derivatives of Sine & Cosine
Derivative of Square Root Function
Understanding how to differentiate square root functions is essential here. The derivative of √u with respect to u is 1/(2√u). This rule is applied twice in the problem: first to differentiate √t and then to differentiate √(1 + √t), which are nested within the sine function.
Recommended video:
01:32Derivatives of Other Trig Functions Example 1
Related Practice
Textbook Question
In Exercises 41–58, find dy/dt.
y = (1 + cos(2t))⁻⁴
Textbook Question
Derivatives
In Exercises 27–32, find dp/dq.
p = (q sin q) / (q² − 1)
Textbook Question
Finding Derivative Values
In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.
f(u) = 1 − (1/u), u = g(x) = (1 / (1 − x)), x = −1
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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