The hyperbolic functions are cosh(x), coth(x), sinh(x), tanh(x), csch(x) and sech(x). They are similar in many ways to the trigonometric functions, which are also called circular functions. The function sinh(x) = (e^x -- e^(-x))/2 and the function coshx = (e^x + e^(-x))/2. The function tanh(x) = sinh(x)/cosh(x), the function coth(x) = 1/tanh(x), the function sech(x) = 1/cosh(x) and the function csch(x) = 1/sinh(x). Taking the derivative of sinh(x) = (e^x -- e^(-x))/2 gives (e^x -- (-e^(-x)))/2 which is the same as (e^x + e^(-x))/2. (e^x + e^(-x))/2 is cosh(x).
Instructions
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1
Determine which hyperbolic function or functions are in the equation. For example, given the function y = cosh(x), cosh(x) is the hyperbolic function.
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2
Take the derivative of the hyperbolic function. The derivative of sinh(x) = cosh(x), the derivative of cosh(x) is sinh(x), the derivative of tanh(x) = sech^2(x), the derivative of coth(x) = -csch^2(x), the derivative of sech(x) = -sech(x)*tanh(x), and the derivative of csch(x) = -csch(x)*coth(x). In the example, d/dx cosh(x) = sinh(x).
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3
Simplify, if possible. In the example, sinh(x) cannot be simplified further.