Textbook Question
If x²y³ = 4/27 and dy/dt = ¹/₂, then what is dx/dt when x = 2?
195
views
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 3.6.52
In Exercises 41–58, find dy/dt.
y = (1/6)(1 + cos²(7t))³
Verified step by step guidance1
Identify the function y = \( \frac{1}{6} (1 + \cos^2(7t))^3 \) and recognize that you need to find \( \frac{dy}{dt} \). This involves using the chain rule for differentiation.
Apply the chain rule: If \( y = u^3 \) where \( u = 1 + \cos^2(7t) \), then \( \frac{dy}{dt} = 3u^2 \cdot \frac{du}{dt} \).
Differentiate \( u = 1 + \cos^2(7t) \) with respect to t. Use the chain rule again: \( \frac{du}{dt} = 2\cos(7t) \cdot (-\sin(7t)) \cdot 7 \).
Simplify \( \frac{du}{dt} \) to \( -14\cos(7t)\sin(7t) \). This can be further simplified using the double angle identity: \( \sin(2x) = 2\sin(x)\cos(x) \), so \( \frac{du}{dt} = -7\sin(14t) \).
Substitute \( u \) and \( \frac{du}{dt} \) back into the expression for \( \frac{dy}{dt} \): \( \frac{dy}{dt} = \frac{1}{6} \cdot 3(1 + \cos^2(7t))^2 \cdot (-7\sin(14t)) \). Simplify this expression to find the derivative.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
8mWas this helpful?
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 the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In this problem, the chain rule helps differentiate the nested functions within y = (1/6)(1 + cos²(7t))³, particularly the inner function cos²(7t) and its outer cube.
Recommended video:
05:02
Intro to the Chain Rule
Derivative of Trigonometric Functions
Understanding how to differentiate trigonometric functions is crucial for solving this problem. The derivative of cos(x) is -sin(x), and when dealing with cos²(x), the power rule and chain rule are applied. Specifically, for cos²(7t), you need to differentiate using the chain rule, considering the inner function 7t and the outer square.
Recommended video:
6:04Introduction to Trigonometric Functions
Power Rule
The power rule is used to differentiate functions of the form x^n, where the derivative is n*x^(n-1). In this problem, the power rule is applied to the function (1 + cos²(7t))³, where the exponent 3 is brought down, and the function inside is raised to the power of 2, followed by differentiating the inner function using the chain rule.
Recommended video:
Guided course
5:50Power Rules
Related Practice
Textbook Question
Find the derivatives of the functions in Exercises 1–42.
𝓻 = ( sin θ )²
( cos θ - 1 )
358
views
Textbook Question
In Exercises 53 and 54, find dr/ds.
r cos 2s + sin²s = π
209
views
Textbook Question
For what value of c is the curve y = c/ (x + 1) tangent to the line through the points (0, 3) and (5, -2)?
174
views
Textbook Question
Find the derivatives of the functions in Exercises 1–42.
𝔂 = 2 tan² x - sec² x
333
views
Textbook Question
A melting ice layer A spherical iron ball 8 in. in diameter is coated with a layer of ice of uniform thickness. If the ice melts at the rate of 10 in³/min, how fast is the thickness of the ice decreasing when it is 2 in. thick? How fast is the outer surface area of ice decreasing?
215
views