Table 2.9.1 lists the definitions of the six hyperbolic functions and their derivatives. As in Section 2.6, there are two cases where Maple's derivatives differ from what is found in the typical calculus text.

As for the derivatives of the trig functions, Maple returns the derivatives of the hyperbolic tangent and cotangent functions in a form that differs from the form found in the typical calculus textbook. (Note the red cells in Table 2.9.1.) This is because early-on, the Maple programmers opined that for these two functions, returning the derivative in terms of the same function was somehow "simpler" than returning it in terms of a different function. Table 2.9.2 contains Maple code for modifying the differentiation rules for these two functions. (The restart is in deference to any remember-table issues.)

If the differentiation rules are arranged as in Table 2.9.3, certain relations between all the hyperbolic functions and their derivatives can be observed.

If each function in the left-hand column is replaced by its co-function, and a minus sign inserted for just the derivative of the hyperbolic cotangent, the right-hand column results.

Table 2.9.4 compares the derivatives of the trig and hyperbolic functions.

Table 2.9.5 contains graphs of the six hyperbolic functions. The graphs in the second row are graphs of the reciprocals of the functions graphed in the first row. Thus, the graph below that of $\sinh \left(x\right)$ is the graph of $\mathrm{csch}\left(x\right)$.

From their graphs, infer that $\sinh \left(x\right)$ is an odd function; and $\cosh \left(x\right)$, and even function. Note also that $\cosh \left(x\right)\ge 1$ for all real $x$.

Table 2.9.6 lists some identities for the hyperbolic functions, and compares them to their trigonometric counterparts.

Figure 2.9.1 suggests why the trig functions are also called the circular functions. Similarly, Figure 2.9.2 suggests why the hyperbolic functions have their name.

The circular functions can be defined in terms of the sine and cosine functions, which themselves satisfy the identity ${\cos }^2\left(t\right)\+{\sin }^2\left(t\right)\=1$. The parametric curve defined by $x\=\cos \left(t\right)\,y\=\sin \left(t\right)$ is therefore the unit circle.

Similarly, the hyperbolic functions can be defined in terms of the hyperbolic sine and cosine functions, which themselves satisfy the identity ${\cosh }^2\left(x\right)-{\sinh }^2\left(x\right)\=1$. The parametric curve defined by $x\=\cosh \left(t\right)\,y\=\sinh \left(t\right)$ is therefore the hyperbola shown in Figure 2.9.2.