Question

Simplify the expressions. $$\sinh (\ln t)$$

Name: Problem 16, Chapter 3: Calculus Single Variable
Uploaded: 2021-04-27T07:36:43-08:00
Duration: 1 min 9 s
Channel: Tyler Moulton
Description: Simplify the expressions. sinh(lnt)

   Simplify the expressions.
$$\sinh (\ln t)$$

Calculus Single Variable

Deborah… 7th Edition

Chapter 3, Problem 16

Instant Answer

Solved by Expert Tyler Moulton

04/27/2021

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Simplify the expressions. $$\sinh (\ln t)$$

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Key Concepts

Function Composition and Simplification

This concept involves recognizing the structure of a composed function and applying the definitions of the inner and outer functions accordingly. It is central to simplifying expressions where functions are nested, such as applying the exponential-logarithmic identity within the hyperbolic sine function.

Hyperbolic Sine Function

The hyperbolic sine function is defined as sinh(x) = (e^x - e^(–x)) / 2. This definition is essential in converting expressions involving hyperbolic functions into forms that use exponential functions, which are often easier to manipulate algebraically.

Exponential and Logarithmic Relationship

Exponential and logarithmic functions are inverses of each other, meaning that e^(ln t) = t. This relationship is key when simplifying expressions in which a logarithm is used as an argument to an exponential function, enabling the direct substitution and simplification of the expression.

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