# Differentiation and integration

## Differentiation of Hyperbolic Functions

• The derivative of sinh(x) is cosh(x).
• The derivative of cosh(x) is sinh(x).
• The derivative of tanh(x) is sech^2(x) where sech(x) is the hyperbolic secant defined as 1/cosh(x).
• Chain rule from calculus applies to all hyperbolic functions. For instance, the derivative of the composite function where f is a function of g is given by f’(g)*g’.

## Integration of Hyperbolic Functions

• The integral of sinh(x) with respect to x is cosh(x) + C, where C is the constant of integration.
• The integral of cosh(x) with respect to x is sinh(x) + C.
•  The integral of tanh(x) with respect to x is ln cosh(x) + C.
• Integration by substitution or parts is often useful when evaluating integrals involving hyperbolic functions.
• The second derivative of sinh(x) is again sinh(x), and the second derivative of cosh(x) is again cosh(x). In contrast to the cyclic nature of the second derivatives of sine and cosine, these show the even and odd function properties.
• Hyperbolic identities can be used to simplify complex expressions before differentiation or integration.
• The fundamental theorem of calculus applies to hyperbolic functions, meaning that the exact area under a curve (the definite integral) may be calculated using antiderivatives.

## Sample Problems for Differentiation and Integration

• Differentiate the function f(x) = 3sinh(2x) + 5x^2cosh(3x).
• Find the integral of the function g(x) = 4cosh(3x) - 1/(2)tanh^2(x).
• Determine the integral of sinh(2x)cosh(x) dx by employing an appropriate substitution or a suitable identity.

Remember, consistency in practice and problem-solving is essential for mastering the differentiation and integration of these functions, just as is the case with trigonometric functions.