# Differentiation and integration

## 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 lncosh(x) + C. - Integration by substitution or parts is often useful when evaluating integrals involving hyperbolic functions.

## Other Important Properties Related to Differentiation and Integration

- 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.