Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Calculus

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

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1:33 minutes

Problem 7.3.13

Textbook Question

11–15. Identities Prove each identity using the definitions of the hyperbolic functions.

cosh 2x = cosh²x + sinh²x (Hint: Begin with the right side of the equation.)

Verified step by step guidance

Start with the right-hand side of the equation: cosh²x + sinh²x.

Recall the definitions of the hyperbolic functions: cosh(x) = (e^x + e^(-x))/2 and sinh(x) = (e^x - e^(-x))/2.

Square both definitions: cosh²x = [(e^x + e^(-x))/2]² and sinh²x = [(e^x - e^(-x))/2]².

Expand the squares: cosh²x = (e^(2x) + 2 + e^(-2x))/4 and sinh²x = (e^(2x) - 2 + e^(-2x))/4.

Add the two results: cosh²x + sinh²x = [(e^(2x) + 2 + e^(-2x)) + (e^(2x) - 2 + e^(-2x))]/4 = (2e^(2x) + 2e^(-2x))/4 = (e^(2x) + e^(-2x))/2, which is the definition of cosh(2x).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hyperbolic Functions

Hyperbolic functions, such as sinh(x) and cosh(x), are analogs of the trigonometric functions but are based on hyperbolas instead of circles. They are defined as sinh(x) = (e^x - e^(-x))/2 and cosh(x) = (e^x + e^(-x))/2. Understanding these definitions is crucial for manipulating and proving identities involving hyperbolic functions.

Hyperbolic Identities

Hyperbolic identities are equations that hold true for hyperbolic functions, similar to trigonometric identities. One fundamental identity is cosh²(x) - sinh²(x) = 1. Recognizing and applying these identities is essential for proving new identities, such as the one in the question.

Proof Techniques

Proof techniques in calculus often involve algebraic manipulation, substitution, and the application of known identities. In this case, starting with the right side of the equation and transforming it to match the left side using hyperbolic definitions and identities is a common strategy. Mastery of these techniques is vital for successfully proving mathematical statements.