Calculus of the Hyperbolic Functions: Learn It 1

  • Differentiate and integrate hyperbolic functions and their inverse forms
  • Understand the practical situations where the catenary curve appears

Derivatives and Integrals of the Hyperbolic Functions

The hyperbolic sine ([latex]sinh[/latex]) and hyperbolic cosine ([latex]cosh[/latex]) functions are defined as:

[latex]\text{sinh}x=\dfrac{{e}^{x}-{e}^{\text{−}x}}{2}\text{     and     }\text{cosh}x=\dfrac{{e}^{x}+{e}^{\text{−}x}}{2}[/latex]

The other hyperbolic functions are then defined in terms of [latex]\text{sinh}x[/latex] and [latex]\text{cosh}x.[/latex] The graphs of these functions provide insights into their behaviors.

This figure has six graphs. The first graph labeled “a” is of the function y=sinh(x). It is an increasing function from the 3rd quadrant, through the origin to the first quadrant. The second graph is labeled “b” and is of the function y=cosh(x). It decreases in the second quadrant to the intercept y=1, then becomes an increasing function. The third graph labeled “c” is of the function y=tanh(x). It is an increasing function from the third quadrant, through the origin, to the first quadrant. The fourth graph is labeled “d” and is of the function y=coth(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis. The fifth graph is labeled “e” and is of the function y=sech(x). It is a curve above the x-axis, increasing in the second quadrant, to the y-axis at y=1 and then decreases. The sixth graph is labeled “f” and is of the function y=csch(x). It has two pieces, one in the third quadrant and one in the first quadrant with a vertical asymptote at the y-axis.
Figure 1. Graphs of the hyperbolic functions.

It is straightforward to develop differentiation formulas for hyperbolic functions. For instance:

[latex]\begin{array}{cc}\hfill \frac{d}{dx}(\text{sinh}x)& =\frac{d}{dx}(\frac{{e}^{x}-{e}^{\text{−}x}}{2})\hfill \\ & =\frac{1}{2}\left[\frac{d}{dx}({e}^{x})-\frac{d}{dx}({e}^{\text{−}x})\right]\hfill \\ & =\frac{1}{2}\left[{e}^{x}+{e}^{\text{−}x}\right]=\text{cosh}x.\hfill \end{array}[/latex]

Similarly, [latex](\frac{d}{dx})\text{cosh}x=\text{sinh}x.[/latex]

derivatives of the hyperbolic functions

Derivatives of the Hyperbolic Functions
[latex]f(x)[/latex] [latex]\frac{d}{dx}f(x)[/latex]
[latex]\text{sinh}x[/latex] [latex]\text{cosh}x[/latex]
[latex]\text{cosh}x[/latex] [latex]\text{sinh}x[/latex]
[latex]\text{tanh}x[/latex] [latex]{\text{sech}}^{2}x[/latex]
[latex]\text{coth}x[/latex] [latex]\text{−}{\text{csch}}^{2}x[/latex]
[latex]\text{sech}x[/latex] [latex]\text{−}\text{sech}x\text{tanh}x[/latex]
[latex]\text{csch}x[/latex] [latex]\text{−}\text{csch}x\text{coth}x[/latex]

The derivatives of hyperbolic functions share similarities with those of trigonometric functions. For example:

[latex](\frac{d}{dx}) \sin x= \cos x[/latex] and [latex](\frac{d}{dx})\text{sinh}x=\text{cosh}x.[/latex]

However, the derivatives of the cosine functions differ in sign:

[latex](\frac{d}{dx}) \cos x=\text{−} \sin x,[/latex] but [latex](\frac{d}{dx})\text{cosh}x=\text{sinh}x.[/latex]

Using the differentiation formulas, we can derive the integral formulas for hyperbolic functions.

 integral formulas for hyperbolic functions

[latex]\begin{array}{cccccccc}\hfill \displaystyle\int \text{sinh}udu& =\hfill & \text{cosh}u+C\hfill & & & \hfill \displaystyle\int {\text{csch}}^{2}udu& =\hfill & \text{−}\text{coth}u+C\hfill \\ \hfill \displaystyle\int \text{cosh}udu& =\hfill & \text{sinh}u+C\hfill & & & \hfill \displaystyle\int \text{sech}u\text{tanh}udu& =\hfill & \text{−}\text{sech}u+C\hfill \\ \hfill \displaystyle\int {\text{sech}}^{2}udu& =\hfill & \text{tanh}u+C\hfill & & & \hfill \displaystyle\int \text{csch}u\text{coth}udu& =\hfill & \text{−}\text{csch}u+C\hfill \end{array}[/latex]

Evaluate the following derivatives:

  1. [latex]\frac{d}{dx}(\text{sinh}({x}^{2}))[/latex]
  2. [latex]\frac{d}{dx}{(\text{cosh}x)}^{2}[/latex]

Evaluate the following integrals:

  1. [latex]\displaystyle\int x\text{cosh}({x}^{2})dx[/latex]
  2. [latex]\displaystyle\int \text{tanh}xdx[/latex]