58.1 Standard Hyperbolic Coefficients

Hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. They appear frequently in various fields of engineering and physics. Their derivatives are straightforward and follow patterns similar to their trigonometric counterparts.

The six basic hyperbolic functions and their derivatives are:

• $$\frac{d}{dx}(\sinh x) = \cosh x$$
• $$\frac{d}{dx}(\cosh x) = \sinh x$$
• $$\frac{d}{dx}(\tanh x) = \text{sech}^2 x$$
• $$\frac{d}{dx}(\coth x) = -\text{csch}^2 x$$
• $$\frac{d}{dx}(\text{sech } x) = -\text{sech } x \tanh x$$
• $$\frac{d}{dx}(\text{csch } x) = -\text{csch } x \coth x$$

58.2 Advanced Hyperbolic Problems

When dealing with more complex hyperbolic functions, such as compositions, products, or quotients, we combine the basic differentiation rules (chain rule, product rule, quotient rule) with the specific derivatives of hyperbolic functions. The chain rule is particularly common here, where $$\frac{d}{dx}(f(u)) = f'(u) \frac{du}{dx}$$.

59.1 Inverse Functions Review

An inverse function "undoes" the action of another function. If $f(a) = b$, then $f^{-1}(b) = a$. For a function to have an inverse, it must be one-to-one (pass the horizontal line test). Graphically, the inverse of a function is a reflection of the original function across the line $y=x$.

To find the inverse of a function $y=f(x)$:

1. Replace $f(x)$ with $y$.
2. Swap $x$ and $y$.
3. Solve the new equation for $y$.
4. Replace $y$ with $f^{-1}(x)$.

The derivative of an inverse function can be found using the formula: $$\frac{d}{dx}(f^{-1}(x)) = \frac{1}{f'(f^{-1}(x))}$$.

59.2 Inverse Trig Differentiation

The inverse trigonometric functions (or arc functions) are the inverse functions of the trigonometric functions. Their derivatives are fundamental in calculus, especially in integration.

• $$\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$$
• $$\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}$$
• $$\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$$
• $$\frac{d}{dx}(\text{arccot } x) = -\frac{1}{1+x^2}$$
• $$\frac{d}{dx}(\text{arcsec } x) = \frac{1}{|x|\sqrt{x^2-1}}$$
• $$\frac{d}{dx}(\text{arccsc } x) = -\frac{1}{|x|\sqrt{x^2-1}}$$

When composed with other functions, remember to apply the chain rule. For example, $$\frac{d}{dx}(\arcsin u) = \frac{1}{\sqrt{1-u^2}} \frac{du}{dx}$$.

59.3 Logarithmic Forms (Inverse Hyperbolic)

Just as inverse trigonometric functions can be defined using angles in a circle, inverse hyperbolic functions can be expressed in terms of natural logarithms. This connection is useful for both understanding their properties and sometimes for integration.

• $$\text{arsinh } x = \ln(x + \sqrt{x^2+1})$$
• $$\text{arcosh } x = \ln(x + \sqrt{x^2-1})$$, for $x \ge 1$
• $$\text{artanh } x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$$, for $|x| < 1$
• $$\text{arcoth } x = \frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)$$, for $|x| > 1$
• $$\text{arsech } x = \ln\left(\frac{1+\sqrt{1-x^2}}{x}\right)$$, for $0 < x \le 1$
• $$\text{arcsch } x = \ln\left(\frac{1}{x} + \frac{\sqrt{1+x^2}}{|x|}\right)$$, for $x \ne 0$

59.4 Inverse Hyperbolic Differentiation

The derivatives of inverse hyperbolic functions also have standard forms, often derived from their logarithmic expressions or by implicit differentiation of the original hyperbolic function.

• $$\frac{d}{dx}(\text{arsinh } x) = \frac{1}{\sqrt{1+x^2}}$$
• $$\frac{d}{dx}(\text{arcosh } x) = \frac{1}{\sqrt{x^2-1}}$$, for $x > 1$
• $$\frac{d}{dx}(\text{artanh } x) = \frac{1}{1-x^2}$$, for $|x| < 1$
• $$\frac{d}{dx}(\text{arcoth } x) = \frac{1}{1-x^2}$$, for $|x| > 1$
• $$\frac{d}{dx}(\text{arsech } x) = -\frac{1}{x\sqrt{1-x^2}}$$, for $0 < x < 1$
• $$\frac{d}{dx}(\text{arcsch } x) = -\frac{1}{|x|\sqrt{1+x^2}}$$, for $x \ne 0$

Again, apply the chain rule when the argument is a function of $x$.