Question

Derive the following derivative formulas given that \(d / d x(\cosh x)=\sinh x\) and \(d / d x(\sinh x)=\cosh x\). \(d / d x(\operatorname{coth} x)=-\operatorname{csch}^{2} x\)

Short answer

Answer: The derivative of \(\coth x\) is \(-\operatorname{csch}^2 x\).

Step-by-step solution

Recall the Quotient Rule

The Quotient Rule states that the derivative of a quotient of two functions is given by: \(\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}\)

Apply the Quotient Rule to \(\coth x\)

Following the Quotient Rule, let \(u(x) = \cosh x\) and \(v(x) = \sinh x\). Thus, we have: \(\frac{d}{dx}(\coth x) = \frac{d}{dx}\left(\frac{\cosh x}{\sinh x}\right) = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}\)

Substitute the given formulas for the derivatives

Now, remember the formulas for the derivatives of \(\cosh x\) and \(\sinh x\): \(d / d x(\cosh x)=\sinh x\) and \(d / d x(\sinh x)=\cosh x\). Replace \(u'(x)\) with \(\sinh x\) and \(v'(x)\) with \(\cosh x\) in the expression above. \(\frac{d}{dx}(\coth x) = \frac{(\sinh x)(\sinh x) - (\cosh x)(\cosh x)}{[\sinh x]^2}\)

Simplify the expression

Now, we have to simplify the expression. We will use the identity \(\cosh^2 x - \sinh^2 x = 1\). Thus, the expression becomes: \(\frac{d}{dx}(\coth x) = \frac{1 - \cosh^2 x}{\sinh^2 x}\)

Rewrite using the definition of \(\operatorname{csch}\)

Finally, we can rewrite the expression using the definition of \(\operatorname{csch}\), which is \(\operatorname{csch} x = \frac{1}{\sinh x}\). Our expression becomes: \(\frac{d}{dx}(\coth x) = -\operatorname{csch}^2 x\) Thus, we have derived the formula for the derivative of \(\coth x\): \(d / d x(\coth x)=-\operatorname{csch}^2 x\).

Key concepts

Derivatives

In calculus, a derivative represents the rate at which a function is changing at any given point. It is a fundamental concept that shows how one quantity changes with respect to another. In simple terms, if you think of a graph, the derivative at a point is the slope of the tangent line at that point.

Understanding derivatives is crucial because they can help solve real-world problems such as predicting population growth or determining the optimum speed for efficiency. When dealing with functions, especially more complex ones like hyperbolic functions, derivatives allow us to explore these ideas further and analyze behavior.

• Derivatives express rates of change.
• They help determine slope and behavior of functions.
• They are used in various fields for optimization and prediction.

Hyperbolic Functions

Hyperbolic functions are similar to trigonometric functions but are based on hyperbolas instead of circles. The main hyperbolic functions are sinh (hyperbolic sine), cosh (hyperbolic cosine), and tanh (hyperbolic tangent). They are defined with exponential functions and appear in many mathematical contexts, especially in calculus.

With hyperbolic functions, it's important to know their derivatives:

• The derivative of \(\cosh x\) is \(\sinh x\).
• The derivative of \(\sinh x\) is \(\cosh x\).

These derivatives are highly useful in solving calculus problems that involve hyperbolic functions. They have unique properties and applications in engineering, physics, and hyperbolic geometry.

Hyperbolic functions offer an alternative to trigonometric functions in certain scenarios, where their geometric interpretation based on hyperbolas can be more suitable.

• Hyperbolic functions are derived from hyperbolas.
• They include sinh, cosh, and tanh.
• Commonly used in advanced calculus and applied sciences.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the quotient of two other functions. It's especially useful when dealing with ratios or divisions in calculus. The rule states that if you have a function \(\frac{u(x)}{v(x)}\), then its derivative can be found using:

\[\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}\]
This formula is key when working specifically with expressions like \(\coth x\), as it involves a division of hyperbolic sine and cosine functions.
When applying the quotient rule:

• Select \(u(x)\) and \(v(x)\) appropriately based on the function's structure.
• Compute \(u'(x)\) and \(v'(x)\).
• Ensure to substitute these derivatives correctly into the formula.

For example, in deriving \(\frac{d}{dx}(\coth x)\), the quotient rule helps simplify and solve for the derivative efficiently by organizing complex expressions into manageable steps.

• The quotient rule is vital for derivatives involving division.
• It requires clear differentiation of the numerator and denominator functions.
• Important for simplifying complex calculus problems.