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Chapter 37: Problem 1

Determine the differential coefficient of: (a) th \(x\) (b) \(\operatorname{sech} x\)

Short Answer

Expert verified
(a) 0, (b) -\operatorname{sech} x \tanh x

Step by step solution

01

Differentiate the Constant Function

The function given is a constant function denoted by \(a\). The derivative of a constant function \(f(x) = c\) is zero because \ \(\frac{d}{dx}c = 0\)\ Therefore, the differential coefficient of \(a\) with respect to \(x\) is \(0\).
02

Use the Chain Rule for \( \operatorname{sech} x\)

The hyperbolic secant function can be expressed as \( \operatorname{sech} x = \frac{1}{\cosh x} \). To differentiate \(\operatorname{sech} x\), we use the chain rule and the quotient rule. Start with the expression \( \operatorname{sech} x = (\cosh x)^{-1} \).
03

Differentiate \((\cosh x)^{-1}\)

Using the chain rule, differentiate \((\cosh x)^{-1}\) with respect to \(x\): \ \( \frac{d}{dx} (\cosh x)^{-1} = -1 \cdot (\cosh x)^{-2} \cdot \sinh x\) \ Simplify to get: \ \( -\frac{\sinh x}{\cosh^2 x}\).
04

Simplify Using Hyperbolic Identity

Apply the identity \(1 - \tanh^2 x = \operatorname{sech}^2 x\) and note that \(\sinh x / \cosh x = \tanh x\). Thus: \ \( \frac{\sinh x}{\cosh^2 x} = \tanh x \operatorname{sech} x\) \ Thus, the derivative of \(\operatorname{sech} x\) is: \(-\operatorname{sech} x \tanh x\).

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Key Concepts

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

Understanding Derivatives
The concept of derivatives in calculus is fundamental. A derivative represents the rate at which a function changes as its input changes. Think of it as the "slope" of a function at any given point. Here are some key points about derivatives:
  • The derivative of a constant function is zero. This is because constants do not change, which means their rate of change is zero.
  • For example, if you have a function \( f(x) = c \), where \( c \) is a constant, then \( \frac{d}{dx}c = 0 \).
Derivatives are powerful tools that help us understand the behavior of functions in various contexts, from physics to engineering.
Exploring the Chain Rule
When dealing with complex functions, the chain rule is essential. The chain rule allows us to differentiate a function that is composed of other functions. Here's how it works:
  • Consider a composite function \( f(g(x)) \). To find its derivative, we use the chain rule formula: \( \frac{df}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx} \).
  • This approach is useful when differentiating functions like \( \operatorname{sech} x = (\cosh x)^{-1} \).
By implementing the chain rule, you can break down complex differentiations into simpler parts, making it easier to understand and solve complex problems.
Understanding Hyperbolic Functions
Hyperbolic functions are analogs of trigonometric functions but for hyperbolas. They have unique properties and identities:
  • The hyperbolic secant function, \( \operatorname{sech} x \), is defined as \( \operatorname{sech} x = \frac{1}{\cosh x} \).
  • To differentiate \( \operatorname{sech} x \), we use both the chain rule and hyperbolic identities, such as \( 1 - \tanh^2 x = \operatorname{sech}^2 x \).
  • Understanding these identities helps simplify the differentiation process, as shown when simplifying \( \frac{\sinh x}{\cosh^2 x} \) to arrive at the derivative \( -\operatorname{sech} x \tanh x \).
Hyperbolic functions are widely used in many advanced fields, including physics and engineering, due to their unique properties and applications.

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Most popular questions from this chapter

Differentiate the following with respect to \(x\) : (a) \(y=4 \operatorname{sh} 2 x-\frac{3}{7} \operatorname{ch} 3 x\) (b) \(y=5\) th \(\frac{x}{2}-2 \operatorname{coth} 4 x\)

Show that the differential coefficient of $$ y=\frac{3 x^{2}}{\operatorname{ch} 4 x} \text { is: } 6 x \operatorname{sech} 4 x(1-2 x \operatorname{th} 4 x) $$

Differentiate the following with respect to the variable: (a) \(y=4 \sin 3 t \operatorname{ch} 4 t\) (b) \(y=\ln (\operatorname{sh} 3 \theta)-4 \operatorname{ch}^{2} 3 \theta\)

Determine \(\frac{\mathrm{d} y}{\mathrm{~d} \theta}\) given (a) \(y=\operatorname{cosech} \theta\) (b) \(y=\operatorname{coth} \theta\)
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