91Ó°ÊÓ

When it comes to differentiating complex functions, the chain rule is an essential tool in calculus. It allows us to find the derivative of a composite function. In simple terms, if you have a function that's made up of other functions, the chain rule helps to untangle this and find the rate at which it changes.

The chain rule states that if you have two functions, say, u(x) and v(u), and you want to find the derivative of the composite function v(u(x)), then the derivative is given by the product of the derivative of the outer function at u and the derivative of the inner function at x. In mathematical notation, this is written as \( \frac{dv}{dx} = \frac{dv}{du} \cdot \frac{du}{dx} \).

This rule was applied to the given exercise in order to differentiate the logarithmic function of a hyperbolic function, by treating the inner hyperbolic function as u and the outer natural logarithm as v.

Hyperbolic functions are analogs of the trigonometric functions but for a hyperbola rather than a circle. They have a wide range of applications in physics, engineering, and mathematics. The most common hyperbolic functions include the hyperbolic sine sinh(x), hyperbolic cosine cosh(x), and hyperbolic tangent tanh(x).

They are defined through the exponential function: \(\sinh(x) = \frac{e^{x} - e^{-x}}{2}\), \(\cosh(x) = \frac{e^{x} + e^{-x}}{2}\), and \(\tanh(x) = \frac{\sinh(x)}{\cosh(x)}\).

These functions are important when dealing with the derivatives of functions involving hyperbolas, such as the function in the given exercise where \(\sech(x)\), another hyperbolic function, appears.

The natural logarithm, denoted as \(\ln(x)\), is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. The derivative of the natural logarithm of a function is given by \(\frac{d}{dx} \ln(u(x)) = \frac{1}{u(x)} \cdot u'(x)\).

This derivative tells us that the rate of change of the natural logarithm of a function with respect to \(x\) is the reciprocal of the function multiplied by the derivative of the function itself. This rule was used in step one of the given exercise where the derivative of the natural logarithm of a \(\sech\) function is needed.

The hyperbolic secant function, \(\sech(x)\), is one of the less common hyperbolic functions but it follows similar rules for differentiation as its counterparts. It is defined as the reciprocal of the hyperbolic cosine: \(\sech(x) = \frac{1}{\cosh(x)}\).

To find the derivative of \(\sech(x)\), one can use the quotient rule or recognize it as a composition of functions and use the chain rule. The derivative of \(\sech(x)\) with respect to \(x\) is \(\sech'(x) = -\sech(x)\tanh(x)\). This was applied in step two of the exercise, 'differentiating the function \(2x\)' by identifying \(\sech(2x)\) as a function of \(x\) composed with \(\sech(u)\) and leveraging the chain rule.