Derivative of Hyperbolic Trig Functions

We now understand the derivatives of hyperbolic trig functions. For instance, the derivative of sinh(x) is cosh(x), and the derivative of cosh(x) is sinh(x). The derivative of tanh(x), often represented as d/dx(tanh x) or (tanh x)’, is sech²x. Each hyperbolic function has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: - Hyperbolic Sine: sinh(x) = (e^x - e^(-x))/2 - Hyperbolic Cosine: cosh(x) = (e^x + e^(-x))/2 - Hyperbolic Tangent: tanh(x) = sinh(x)/cosh(x) - Hyperbolic Secant: sech(x) = 1/cosh(x)

The derivatives of hyperbolic trig functions can be denoted as follows: - d/dx(sinh x) = cosh x - d/dx(cosh x) = sinh x - d/dx(tanh x) = sech²x These formulas apply to all x in the domain of the respective hyperbolic functions.

We can derive the derivatives of hyperbolic trig functions using mathematical proofs. To show this, we will use exponential definitions along with differentiation rules. There are several methods to prove these, such as: - By First Principle - Using Chain Rule We will now demonstrate that the differentiation of tanh x results in sech²x using these methods: By First Principle The derivative of tanh x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of tanh x using the first principle, we consider f(x) = tanh x. Its derivative can be expressed as the following limit. f'(x) = lim(h→0) [f(x + h) - f(x)] / h Using the definition tanh(x) = sinh(x)/cosh(x), and applying the quotient rule: f'(x) = [cosh(x)·d/dx(sinh x) - sinh(x)·d/dx(cosh x)] / cosh²x = [cosh(x)·cosh(x) - sinh(x)·sinh(x)] / cosh²x = 1/cosh²x = sech²x Hence, proved. Using Chain Rule To prove the differentiation of tanh x using the chain rule, we start with the definition: tanh x = sinh x / cosh x Let u = sinh x and v = cosh x. By the quotient rule: d/dx [u/v] = [u'v - uv'] / v² Substituting u = sinh x and v = cosh x: d/dx(tanh x) = [cosh x·cosh x - sinh x·sinh x] / cosh²x = [cosh²x - sinh²x] / cosh²x = 1/cosh²x = sech²x

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a bit complex. To understand them better, consider how the acceleration of a car changes (second derivative) and how the rate of acceleration changes (third derivative). Higher-order derivatives help analyze functions like sinh(x), cosh(x), and tanh(x). For the first derivative of a function, we write f′(x), indicating the rate of change. The second derivative is derived from the first derivative, denoted as f′′(x). Similarly, the third derivative, f′′′(x), is the result of differentiating the second derivative. For the nth derivative of a hyperbolic function, we use fⁿ(x), representing the nth derivative.

- At x = 0, the derivative of sinh(x) is cosh(0) = 1, and the derivative of cosh(x) is sinh(0) = 0. - For tanh(x), when x approaches ±∞, the derivative sech²(x) approaches 0, reflecting the asymptotic behavior of tanh(x).

- Hyperbolic Sine: A function defined as sinh(x) = (e^x - e^(-x))/2. - Hyperbolic Cosine: A function defined as cosh(x) = (e^x + e^(-x))/2. - Hyperbolic Tangent: A function defined as tanh(x) = sinh(x)/cosh(x). - Hyperbolic Secant: A function defined as sech(x) = 1/cosh(x). - Chain Rule: A differentiation rule used to find the derivative of composite functions.