FAC1004 L18 — Hyperbolic Functions (Derivatives & Integrals)
Learning Outcome
To find and apply derivatives and integrals of hyperbolic functions.
Definitions (Recap)
The function $e^x$ can be expressed as the sum of an even function and an odd function:
$$e^x = \frac{e^x + e^{-x}}{2} + \frac{e^x - e^{-x}}{2}$$
-
Even function: hyperbolic cosine of $x$ $$\cosh x = \frac{e^x + e^{-x}}{2}$$
-
Odd function: hyperbolic sine of $x$ $$\sinh x = \frac{e^x - e^{-x}}{2}$$
Derivatives of Hyperbolic Functions
Derivatives for hyperbolic functions can be obtained by expressing the function in terms of $e^x$ and $e^{-x}$.
Derivation of $\frac{d}{dx}\sinh x$
$$\frac{d}{dx}[\sinh x] = \frac{d}{dx}\left[\frac{e^x - e^{-x}}{2}\right] = \frac{e^x + e^{-x}}{2} = \cosh x$$
Derivation of $\frac{d}{dx}\cosh x$
$$\frac{d}{dx}[\cosh x] = \frac{d}{dx}\left[\frac{e^x + e^{-x}}{2}\right] = \frac{e^x - e^{-x}}{2} = \sinh x$$
General Derivative Formulas (Chain Rule)
Let $u$ be a differentiable function of $x$:
| Function | Derivative |
|---|---|
| $\sinh u$ | $\cosh u \cdot \frac{du}{dx}$ |
| $\cosh u$ | $\sinh u \cdot \frac{du}{dx}$ |
| $\tanh u$ | $\text{sech}^2 u \cdot \frac{du}{dx}$ |
| $\coth u$ | $-\text{csch}^2 u \cdot \frac{du}{dx}$ |
| $\text{sech } u$ | $-\text{sech } u \tanh u \cdot \frac{du}{dx}$ |
| $\text{csch } u$ | $-\text{csch } u \coth u \cdot \frac{du}{dx}$ |
Worked Examples
Example 1: Find $\frac{d}{dx}[\cosh(x^3)]$
$$\frac{d}{dx}[\cosh(x^3)] = \sinh(x^3) \cdot \frac{d}{dx}[x^3] = 3x^2 \sinh(x^3)$$
Example 2: Find $\frac{d}{dx}[\ln(\tanh x)]$
$$\frac{d}{dx}[\ln(\tanh x)] = \frac{1}{\tanh x} \cdot \frac{d}{dx}[\tanh x] = \frac{\text{sech}^2 x}{\tanh x}$$
Integrals of Hyperbolic Functions
The following theorem provides a complete list of the generalized integration formulas for hyperbolic functions.
Basic Integration Formulas
| Integral | Result |
|---|---|
| $\int \sinh u , du$ | $\cosh u + C$ |
| $\int \cosh u , du$ | $\sinh u + C$ |
| $\int \text{sech}^2 u , du$ | $\tanh u + C$ |
| $\int \text{csch}^2 u , du$ | $-\coth u + C$ |
| $\int \text{sech } u \tanh u , du$ | $-\text{sech } u + C$ |
| $\int \text{csch } u \coth u , du$ | $-\text{csch } u + C$ |
Worked Examples
Example 1: Evaluate $\int \sinh^5 x \cosh x , dx$
Let $u = \sinh x$, then $du = \cosh x , dx$.
$$\int \sinh^5 x \cosh x , dx = \int u^5 , du = \frac{1}{6}u^6 + C = \frac{1}{6}\sinh^6 x + C$$
Example 2: Evaluate $\int \tanh x , dx$
Rewrite $\tanh x = \frac{\sinh x}{\cosh x}$.
Let $u = \cosh x$, then $du = \sinh x , dx$.
$$\int \tanh x , dx = \int \frac{\sinh x}{\cosh x} , dx = \int \frac{1}{u} , du = \ln u + C = \ln(\cosh x) + C$$
Summary
This lecture develops the calculus of hyperbolic functions. Key takeaways:
- Derivatives of hyperbolic functions follow patterns similar to trigonometric functions but with sign differences (no alternating sign for $\cosh x$).
- The chain rule applies to hyperbolic functions exactly as it does to trigonometric functions.
- Integration formulas are the reverse of differentiation formulas.
- Substitution techniques (u-substitution) are essential for evaluating hyperbolic integrals.
Related
- FAC1004 - Advanced Mathematics II (Computing) — main course page
- Hyperbolic Functions — concept page
- FAC1004 L17 — Hyperbolic Functions — introduction lecture
- FAC1004 L19-L20 — Inverse Hyperbolic Functions — next lecture
Source File
LECTURE_NOTES_2526/L18 Hyperbolic Function full version.pdf