# Section 1.2 Calculus 2

Hyperbolic Functions

## 1.2 Hyperbolic Functions

### 1.2.1 Hyperbolic Sine and Cosine

• Exponential functions are used to define the hyperbolic functions, which behave like trigonometric functions in many ways.
• $$\sinh x = \frac{e^x-e^{-x}}{2}$$
• $$\cosh x = \frac{e^x+e^{-x}}{2}$$
• Example Evaluate $$\sinh(0)$$ and $$\cosh(0)$$.
• Example Evaluate $$\cosh(\ln 4)$$.
• Example Prove that $$\sinh(2x)=2\sinh(x)\cosh(x)$$.

$$\newcommand{\sech}{\mathrm{sech}\,}$$ $$\newcommand{\csch}{\mathrm{csch}\,}$$

### 1.2.2 Other Hyperbolic Functions

• The other hypberbolic functions are defined the same way their trig counterparts are, and have similar properties.
• $$\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x-e^{-x}}{e^x+e^{-x}}$$
• $$\coth x = \frac{\cosh x}{\sinh x} = \frac{e^x+e^{-x}}{e^x-e^{-x}}$$
• $$\sech x = \frac{1}{\cosh x}=\frac{2}{e^x+e^{-x}}$$
• $$\csch x = \frac{1}{\sinh x}=\frac{2}{e^x-e^{-x}}$$
• Example Evaluate $$\sech(-\ln 2)$$.
• Example Prove that $$\tanh^2(x)=1-\sech^2(x)$$.

### 1.2.3 Derivatives and Integrals of Hyperbolic Functions

• Their derivatives also behave similarly.
• $$\frac{d}{dx}[\sinh x] = \cosh x$$
• $$\frac{d}{dx}[\cosh x] = \sinh x$$
• $$\frac{d}{dx}[\tanh x] = \sech^2 x$$
• $$\frac{d}{dx}[\coth x] = -\csch^2 x$$
• This is correct, but the video incorrectly leaves off the negative
• $$\frac{d}{dx}[\sech x] = -\sech x\tanh x$$
• $$\frac{d}{dx}[\csch x] = -\csch x\coth x$$
• Example Use their definitions to prove that $$\frac{d}{dx}[\cosh x]=\sinh x$$.
• Example Use their definitions to prove that $$\frac{d}{dx}[\coth x]=\csch^2 x$$.
• Example Compute $$\frac{d}{dx}[\sinh(2x)+\coth(x^2)]$$.
• Their integral formulas may be found by just reversing the equations.
• $$\int \cosh x\,dx = \sinh x + C$$
• $$\int \sinh x\,dx = \cosh x + C$$
• $$\int \sech^2 x\,dx = \tanh x + C$$
• $$\int \csch^2 x\,dx = -\coth x + C$$
• $$\int\sech x\tanh x\,dx = -\sech x + C$$
• $$\int\csch x\coth x\,dx = -\csch x + C$$
• Example Find $$\int (4\csch^2 x-3\sinh x)\,dx$$.

### Review Exercises

1. Evaluate $$\sinh(\ln 6)$$.
2. Prove that $$\cosh (2x) = \cosh^2 x + \sinh^2 x$$.
3. Prove that $$\cosh^2 x - \sinh^2 x = 1$$.
4. Evaluate $$\tanh(\ln 3)$$.
5. Simplify $$\sinh(x)\coth(x)\cosh(x)-\frac{1}{\csch^2(x)}$$. (Hint: convert everything to $$\sinh x$$ and $$\cosh x$$.)
6. Prove that $$\frac{d}{dx}[\sinh x] = \cosh x$$.
7. Prove that $$\frac{d}{dx}[\sech x] = -\sech x\tanh x$$. (Hint: use the fact that $$\frac{d}{dx}[\cosh x] = \sinh x$$.)
8. Compute $$\frac{d}{dx}[\tanh(3x)-\sech(\ln x)]$$.
9. Find $$\int (3\csch x\coth x - 2\sinh x)\,dx$$.
10. Let $$\sinh^{\leftarrow}(x)$$ be the inverse function of $$\sinh(x)$$. Use the facts $$\frac{d}{dx}[f^{\leftarrow}(x)]=\frac{1}{f’(f^{\leftarrow}(x))}$$ and $$\cosh^2 x-\sinh^2 x = 1$$ to prove that $$\frac{d}{dx}[\sinh^{\leftarrow}(x)]=\frac{1}{\sqrt{1+x^2}}$$.
11. Prove that $$\sinh^{\leftarrow}(x)=\ln(\sqrt{x^2+1}+x)$$.
12. Evaluate $$\cosh(\ln 2)$$.
13. Differentiate $$f(x)=\tanh(x^2)-\cosh(2x+1)$$.

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 7.3