Domanda del libro di testo
Derivative Calculations
In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).
y = cos u, u = −x/3
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Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 3, Problema 3.23
Find the derivatives of the functions in Exercises 1–42.
𝔂 = x⁻¹/² sec (2x)²
Guida verificata passo dopo passo1
Identify the function components: The given function is 𝔂 = x-1/2 sec((2x)2). This is a product of two functions: u(x) = x-1/2 and v(x) = sec((2x)2).
Apply the product rule: The derivative of a product of two functions u(x) and v(x) is given by (uv)' = u'v + uv'.
Differentiate u(x): The derivative of u(x) = x-1/2 is u'(x) = -1/2 * x-3/2.
Differentiate v(x): To find v'(x), use the chain rule. The derivative of sec(z) is sec(z)tan(z), where z = (2x)2. First, find the derivative of z with respect to x, which is dz/dx = 4x. Then, v'(x) = sec((2x)2)tan((2x)2) * 4x.
Combine the results: Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula to find the derivative of 𝔂. Simplify the expression to obtain the final derivative.
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Derivative Rules
Understanding the rules of differentiation, such as the product rule, quotient rule, and chain rule, is essential for finding derivatives of complex functions. The product rule is used when differentiating products of functions, while the chain rule is necessary for composite functions. Mastery of these rules allows for systematic and accurate differentiation.
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5:50
Power Rules
Trigonometric Functions
The function sec(2x) is a trigonometric function, specifically the secant function, which is the reciprocal of the cosine function. Knowing the derivatives of trigonometric functions, such as sec(x), is crucial for differentiating expressions involving them. The derivative of sec(x) is sec(x)tan(x), which will be applied in this context.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Power Rule
The power rule is a fundamental concept in calculus that states if f(x) = x^n, then f'(x) = n*x^(n-1). This rule is particularly useful for differentiating functions with exponents, such as x^(-1/2) in the given function. Applying the power rule correctly is vital for simplifying the differentiation process.
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Find the derivatives of the functions in Exercises 1–42.
𝔂 = (x + 1)² (x² + 2x)
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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.
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