10–19. Derivatives Find the derivatives of the following functions.
f(x) = (sinh x) / (1 + sinh x)

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Identify the function to differentiate: \(f(x) = \frac{\sinh x}{1 + \sinh x}\), which is a quotient of two functions.

Recall the Quotient Rule for derivatives: if \(f(x) = \frac{u(x)}{v(x)}\), then \(f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\).

Set \(u(x) = \sinh x\) and \(v(x) = 1 + \sinh x\). Compute their derivatives: \(u'(x) = \cosh x\) and \(v'(x) = \cosh x\).

Apply the Quotient Rule: substitute \(u\), \(v\), \(u'\), and \(v'\) into the formula to get \(f'(x) = \frac{\cosh x (1 + \sinh x) - \sinh x \cosh x}{(1 + \sinh x)^2}\).

Simplify the numerator by factoring and combining like terms to express the derivative in its simplest form.

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Hyperbolic Functions

Hyperbolic functions like sinh(x) are analogs of trigonometric functions but based on hyperbolas. The sinh function is defined as (e^x - e^(-x))/2 and has properties similar to sine, including specific derivatives that are essential for differentiation.

Quotient Rule

The quotient rule is used to differentiate functions expressed as one function divided by another. It states that the derivative of f(x)/g(x) is (f'(x)g(x) - f(x)g'(x)) / [g(x)]^2, which is crucial for finding the derivative of the given function.

Derivative of Hyperbolic Sine

The derivative of sinh(x) is cosh(x), another hyperbolic function defined as (e^x + e^(-x))/2. Knowing this derivative is necessary to apply the quotient rule correctly when differentiating the given function.

10–19. Derivatives Find the derivatives of the following functions.f(x) = (sinh x) / (1 + sinh x)