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Chapter 5: Problem 17

Find the derivative of the function. $$ f(x)=\ln (\sinh x) $$

Short Answer

Expert verified
The derivative of the function \( f(x)=\ln (\sinh x) \) is \( f'(x)= \coth x \).

Step by step solution

01

Identify the function

The function provided is \( f(x)=\ln (\sinh x) \) and we need to find its derivative.
02

Apply chain rule and derivative of natural logarithm

We apply the chain rule as \(\frac{d}{dx}\ln u= \frac{u'}{u}\) where \( u=\sinh x \). This gives us \( f'(x)= \frac{\sinh' x}{\sinh x} \).
03

Derivative of hyperbolic sine function

Apply the rule for the derivative of the hyperbolic sine function, \( \sinh' x = \cosh x \). This gives us \( f'(x)= \frac{\cosh x}{\sinh x} \).
04

Simplification

The ratio \(\frac{\cosh x}{\sinh x}\) simplifies to \( \coth x \), hence \( f'(x)= \coth x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative
In mathematics, a derivative represents the rate at which a function is changing at a given point. It is a fundamental concept in calculus, often introduced as the slope of the tangent line to the graph of the function. The derivative is denoted by \( f'(x) \) or \( \frac{df}{dx} \).
  • The derivative of a function provides insight into the function's behavior, such as identifying local maxima and minima.
  • In a broader sense, the derivative also measures how quantities change in relation to each other.
  • This is especially useful in various applications, including physics, engineering, and economics, where relationships between changing quantities are analyzed.
When calculating the derivative, we utilize derivative rules and techniques like the power rule, product rule, quotient rule, and chain rule. Each of these rules has its specific uses based on the structure of the function being differentiated.
Chain Rule
The chain rule is a derivative rule used when computing the derivative of a composite function. A composite function is formed when one function is applied inside another. The chain rule provides an efficient way to differentiate such functions.
Formulaically, if we have a function \( f(x) = g(h(x)) \), then the chain rule states:\[\frac{df}{dx} = g'(h(x)) \cdot h'(x)\]
  • This means that the derivative of the outer function \( g \) (evaluated at the inner function \( h(x) \)) is multiplied by the derivative of the inner function \( h \).
  • In the given exercise, the chain rule is applied to the function \( f(x)=\ln(\sinh x) \), where \( u = \sinh x \).
  • The derivative involves taking \( \frac{d}{dx} \ln(u) = \frac{u'}{u} \), whereby \( u = \sinh x \), leading to \( u' = \cosh x \).
Hyperbolic Functions
Hyperbolic functions are analogs of trigonometric functions, but unlike trig functions, they are based on hyperbolas rather than circles. The basic hyperbolic functions are \( \sinh \), \( \cosh \), \( \tanh \), along with their reciprocals \( \coth \), \( \sech \), and \( \csch \).
  • For the hyperbolic sine function \( \sinh x \), its formula is \( \sinh x = \frac{e^x - e^{-x}}{2} \).
  • The derivative of \( \sinh x \) is \( \cosh x \), where \( \cosh x = \frac{e^x + e^{-x}}{2} \).
  • Hyperbolic functions are crucial in fields like calculus and differential equations, providing solutions to various types of equations.
  • The simplification of \( \frac{\cosh x}{\sinh x} \) provides \( \coth x \), the hyperbolic cotangent function.
Hyperbolic functions appear in many areas of mathematics, including the solutions of certain differential equations and in the parameterization of hyperboloids.
Natural Logarithm
The natural logarithm, denoted by \( \ln(x) \), is a logarithm to the base \( e \), where \( e \) is an irrational and transcendental constant approximately equal to 2.71828. It is one of the most common logarithms used in calculus and arises in many areas of mathematics and science.
  • The natural logarithm is the inverse of the exponential function, meaning \( \ln(e^x) = x \).
  • It satisfies the important property \( \ln(ab) = \ln(a) + \ln(b) \).
  • The derivative of the natural logarithm is \( \frac{1}{x} \), which plays a crucial role in solving calculus problems involving logarithmic differentiation.
  • In our exercise, \( \ln(\sinh x) \) requires using the chain rule to find the derivative.
The properties of natural logarithms make them invaluable for simplifying calculations and solving logarithmic equations in many scientific contexts.

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Most popular questions from this chapter

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