Textbook Question
In Exercises 41–58, find dy/dt.
y = (t tan(t))¹⁰
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 3.9.18
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = x√(1 − x²)
Verified step by step guidance1
Step 1: Identify the function y = x√(1 − x²). This is a product of two functions: x and √(1 − x²).
Step 2: Apply the product rule for differentiation, which states that if you have a function y = u*v, then dy/dx = u'(v) + u(v'). Here, let u = x and v = √(1 − x²).
Step 3: Differentiate u = x with respect to x, which gives u' = 1.
Step 4: Differentiate v = √(1 − x²) with respect to x. Use the chain rule: v' = (1/2)(1 − x²)^(-1/2) * (-2x). Simplify this expression.
Step 5: Substitute u', v, u, and v' into the product rule formula: dy/dx = u'(v) + u(v'). Simplify the expression to find dy/dx.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output changes as its input changes. It is a fundamental concept in calculus, representing the rate of change or slope of the function at a given point. For a function y = f(x), the derivative is denoted as dy/dx and can be found using various rules such as the power rule, product rule, and chain rule.
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Derivatives
Product Rule
The product rule is used to find the derivative of a product of two functions. If y = u(x)v(x), where both u and v are functions of x, the derivative dy/dx is given by u'(x)v(x) + u(x)v'(x). This rule is essential when dealing with functions that are multiplied together, as in the given problem y = x√(1 − x²).
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05:18The Product Rule
Chain Rule
The chain rule is a method for finding the derivative of a composite function. If a function y = f(g(x)) is composed of two functions, the derivative dy/dx is found by multiplying the derivative of the outer function by the derivative of the inner function: f'(g(x))g'(x). This rule is crucial when differentiating functions like √(1 − x²), where the inner function is 1 − x².
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05:02Intro to the Chain Rule
Related Practice
Textbook Question
For what value or values of the constant m, if any, is
ƒ(x) = { sin 2x, x ≤ 0
{ mx, x > 0
a. continuous at x = 0?
b. differentiable at x = 0?
Give reasons for your answers.
Textbook Question
The eight curve Find the slopes of the curve y⁴ = y² – x² at the two points shown here.
Textbook Question
Differential Estimates of Change
In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
The change in the lateral surface area S = 2πrh of a right circular cylinder when the height changes from h₀ to h₀ + dh and the radius does not change
Textbook Question
Free-Fall Applications
Free fall on Mars and Jupiter The equations for free fall at the surfaces of Mars and Jupiter (s in meters, t in seconds) are s = 1.86t² on Mars and s = 11.44t² on Jupiter. How long does it take a rock falling from rest to reach a velocity of 27.8 m/sec (about 100 km/h) on each planet?
Textbook Question
Derivatives
In Exercises 23–26, find dr/dθ.
r = (1 + sec θ) sin θ