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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 3.10.31
Capitolo 3, Problema 3.10.31

Evaluate the derivative of the following functions.
f(u) = csc-1 (2u + 1)

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1
Recognize that the function f(u) = csc^{-1}(2u + 1) involves the inverse cosecant function. The derivative of csc^{-1}(x) with respect to x is -1 / (|x| * sqrt(x^2 - 1)).
Apply the chain rule to differentiate f(u) = csc^{-1}(2u + 1). The chain rule states that if you have a composite function f(g(u)), the derivative is f'(g(u)) * g'(u).
Identify the inner function g(u) = 2u + 1. Differentiate g(u) with respect to u to find g'(u). The derivative of 2u + 1 is 2.
Substitute g(u) = 2u + 1 into the derivative formula for csc^{-1}(x). This gives us -1 / (|2u + 1| * sqrt((2u + 1)^2 - 1)).
Multiply the result from the previous step by g'(u), which is 2, to apply the chain rule. The final expression for the derivative is -2 / (|2u + 1| * sqrt((2u + 1)^2 - 1)).

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Inverse Trigonometric Functions

Inverse trigonometric functions, such as csc<sup>-1</sup> (x), are the inverses of the standard trigonometric functions. They are used to find angles when given a ratio. Understanding their properties and how they relate to their corresponding functions is crucial for differentiation.
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Derivatives of Other Inverse Trigonometric Functions

Chain Rule

The chain rule is a fundamental differentiation technique used when differentiating composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). This rule is essential for evaluating the derivative of functions like f(u) = csc<sup>-1</sup> (2u + 1), where the inner function is 2u + 1.
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Intro to the Chain Rule

Derivative of Inverse Functions

The derivative of an inverse function can be calculated using the formula: if y = f<sup>-1</sup>(x), then dy/dx = 1/(df/dy). For inverse trigonometric functions, specific derivatives exist, such as the derivative of csc<sup>-1</sup>(x), which is -1/(|x|√(x²-1)). Knowing these derivatives is vital for solving the given problem.
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Derivatives of Inverse Sine & Inverse Cosine
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