Prove the formulas given in this table for the derivatives of the functions cosh, tanh, csch, sech, and coth. Which of the following are proven correctly? (Select all that apply.)
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
Prove the formulas given in this table for the derivatives of the functions cosh, tanh, csch, sech, and coth. Which of the following are proven correctly? (Select all that apply.)
![## Derivatives of Hyperbolic Functions
This page provides a detailed transcription of the derivatives of various hyperbolic functions, using common identities and properties. These can be helpful for students studying calculus or engineering courses. Below, each boxed derivative is broken down step by step.
### 1. Derivative of the Hyperbolic Cotangent Function
\[
\boxed{\frac{d}{dx} (\coth{x}) = \frac{d}{dx} \left(\frac{\sinh{x}}{\cosh{x}}\right) = \frac{\cosh{x} \cdot \cosh{x} - \sinh{x} \cdot \sinh{x}}{\cosh^2{x}} = \frac{\cosh^2{x} - \sinh^2{x}}{\cosh^2{x}} = \frac{1}{\cosh^2{x}} = -\csch^2{x}}
\]
### 2. Derivative of the Hyperbolic Cosecant Function
\[
\boxed{\frac{d}{dx} (\csch{x}) = \frac{d}{dx} \left(\frac{1}{\sinh{x}}\right) = -\frac{\cosh{x}}{\sinh^2{x}} = -\frac{1}{\sinh{x}} \cdot \frac{\cosh{x}}{\sinh{x}} = -\csch{x} \cdot \coth{x}}
\]
### 3. Derivative of the Hyperbolic Cosine Function
\[
\boxed{\frac{d}{dx} (\cosh{x}) = \frac{d}{dx} \left(\frac{1}{2} (e^x+e^{-x})\right) = \frac{1}{2} (e^x - e^{-x}) = \sinh{x}}
\]
### 4. Alternative Derivative of the Hyperbolic Cosecant Function
\[
\boxed{\frac{d}{dx} (\csch{x}) = \frac{d}{dx} \left(\frac{1}{\sinh{x}}\right) = -\frac{\cosh{x}}{\sinh^2{x}} = -\frac{1}{\sinh{x}} \cdot \frac{\cosh{x}}{\sinh{x}} = -\csch{x}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b2dbfc9-a79e-4aca-b831-3a84eb9e1ebd%2F306e9742-53f8-4393-92b4-d2964ae9aef0%2F65hyb_processed.png&w=3840&q=75)
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