Textbook Question
If x²y³ = 4/27 and dy/dt = ¹/₂, then what is dx/dt when x = 2?
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 3.11
Find the derivatives of the functions in Exercises 1–42.
𝔂 = 2 tan² x - sec² x
Verified step by step guidance1
Step 1: Identify the function components. The function given is 𝔂 = 2 tan² x - sec² x. This consists of two terms: 2 tan² x and -sec² x.
Step 2: Apply the chain rule to differentiate 2 tan² x. The chain rule states that if you have a function g(x) = f(u(x)), then g'(x) = f'(u(x)) * u'(x). Here, let u(x) = tan x, so tan² x = (tan x)². Differentiate using the chain rule: d/dx [2(tan x)²] = 2 * 2(tan x) * sec² x.
Step 3: Differentiate -sec² x. The derivative of sec x is sec x tan x, so the derivative of sec² x is 2 sec x * sec x tan x = 2 sec² x tan x. Therefore, d/dx [-sec² x] = -2 sec² x tan x.
Step 4: Combine the derivatives from steps 2 and 3. The derivative of the entire function 𝔂 = 2 tan² x - sec² x is the sum of the derivatives of its individual terms: 4 tan x sec² x - 2 sec² x tan x.
Step 5: Simplify the expression if possible. Notice that both terms have a common factor of sec² x tan x. Factor this out to simplify the expression: (4 tan x - 2) sec² x tan x.
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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 value changes as its input value changes. It is a fundamental concept in calculus that represents the slope of the tangent line to the curve of the function at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and chain rule.
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Derivatives
Trigonometric Functions
Trigonometric functions, such as sine, cosine, tangent, secant, and their inverses, are essential in calculus, especially when dealing with derivatives involving angles. Each function has specific derivatives that can be derived from their definitions or unit circle properties. Understanding these functions and their derivatives is crucial for solving problems involving angles and periodic behavior.
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6:04Introduction to Trigonometric Functions
Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if a function y is defined as a composition of two functions, say u(x) and v(u), then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is particularly useful when differentiating functions like tan²(x) or sec²(x) in the given problem.
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05:02Intro to the Chain Rule
Related Practice
Textbook Question
Derivative of multiples Does knowing that a function g(t) is differentiable at t = 7 tell you anything about the differentiability of the function 3g at t = 7? Give reasons for your answer.
Textbook Question
In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.
g(x) = { 2x − x³ − 1, x ≥ 0
x − (1 / (x + 1)), x < 0
Textbook Question
Find the derivatives of the functions in Exercises 19–40.
y = (4x + 3)⁴(x + 1)⁻³
Textbook Question
The best quantity to order One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is
A(q) = (km / q) + cm + (hq / 2),
where q is the quantity you order when things run low (shoes, TVs, brooms, or whatever the item might be); k is the cost of placing an order (the same, no matter how often you order); c is the cost of one item (a constant); m is the number of items sold each week (a constant); and h is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security).
Find dA/dq and d²A/dq².
Textbook Question
Slopes and Tangent Lines
In Exercises 1–4, use the grid and a straight edge to make a rough estimate of the slope of the curve (in y-units per x-unit) at the points P₁ and P₂.