10–19. Derivatives Find the derivatives of the following functions.

f(t) = cosh t sinh t

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Recall the product rule for derivatives: if you have a function \(f(t) = u(t) v(t)\), then its derivative is \(f'(t) = u'(t) v(t) + u(t) v'(t)\).

Identify the two functions in the product: \(u(t) = \cosh t\) and \(v(t) = \sinh t\).

Find the derivatives of each function separately: \(\frac{d}{dt} \cosh t = \sinh t\) and \(\frac{d}{dt} \sinh t = \cosh t\).

Apply the product rule: \(f'(t) = (\sinh t)(\sinh t) + (\cosh t)(\cosh t)\).

Simplify the expression if possible, using hyperbolic identities such as \(\cosh^2 t - \sinh^2 t = 1\).

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주요 개념

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Hyperbolic Functions

Hyperbolic functions, such as sinh(t) and cosh(t), are analogs of trigonometric functions but based on hyperbolas. They have unique properties and identities, like cosh²(t) - sinh²(t) = 1, which are useful in differentiation and integration.

Product Rule for Differentiation

The product rule states that the derivative of a product of two functions u(t) and v(t) is u'(t)v(t) + u(t)v'(t). This rule is essential when differentiating functions like f(t) = cosh(t) * sinh(t), where both factors depend on t.

Derivatives of Hyperbolic Functions

The derivatives of basic hyperbolic functions are: d/dt[sinh(t)] = cosh(t) and d/dt[cosh(t)] = sinh(t). Knowing these derivatives allows you to apply the product rule correctly to find the derivative of functions involving hyperbolic terms.

10–19. Derivatives Find the derivatives of the following functions.f(t) = cosh t sinh t