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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
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. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.32
Capitolo 7, Problema 7.3.32

22–36. Derivatives Find the derivatives of the following functions.


f(t) = 2 tanh⁻¹ √t

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Step 1: Recognize that the function f(t) = 2 tanh⁻¹(√t) involves the inverse hyperbolic tangent function (tanh⁻¹) and a square root. To differentiate this, we will use the chain rule and the derivative formula for tanh⁻¹(x).
Step 2: Recall the derivative formula for tanh⁻¹(x): d/dx[tanh⁻¹(x)] = 1 / (1 - x²). This will be applied to the argument √t.
Step 3: Apply the chain rule. First, differentiate the outer function 2 tanh⁻¹(√t) with respect to √t, which gives 2 * (1 / (1 - (√t)²)).
Step 4: Next, differentiate the inner function √t with respect to t. Recall that d/dt[√t] = 1 / (2√t). Multiply this result with the derivative from Step 3.
Step 5: Combine the results from Steps 3 and 4 to express the derivative of f(t). The final derivative will be f'(t) = 2 * (1 / (1 - t)) * (1 / (2√t)). Simplify if needed.

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Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the function's graph at any given point. The process of finding a derivative is called differentiation, and it is essential for understanding how functions behave and change.
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Derivatives

Inverse Hyperbolic Functions

The function tanh⁻¹, or inverse hyperbolic tangent, is the inverse of the hyperbolic tangent function. It is used to find the value of the original variable when given a hyperbolic tangent value. Understanding how to differentiate inverse hyperbolic functions is crucial for solving problems involving these functions, as they have specific derivative formulas.
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Asymptotes of Hyperbolas

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function is composed of two or more functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This rule is particularly important when dealing with functions like f(t) = 2 tanh⁻¹(√t), where the argument of the inverse function is itself a function of t.
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Intro to the Chain Rule
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