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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.32
Chapter 7, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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Related Practice
Textbook Question

37–56. Integrals Evaluate each integral.

∫ cosh 2x dx

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tanh(−x) = −tanh x

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Uranium dating Uranium-238 (U-238) has a half-life of 4.5 billion years. Geologists find a rock containing a mixture of U-238 and lead, and they determine that 85% of the original U-238 remains; the other 15% has decayed into lead. How old is the rock?

Textbook Question

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.


∫₀ˡⁿ ² (e^{3x} − e^{−3x}) / (e^{3x} + e^{−3x}) dx

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Arc length Use the result of Exercise 108 to find the arc length of the curve: f(x) = ln |tanh(x / 2)| on [ln 2, ln 8].