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Derivatives of Hyperbolic Functions

Function Derivative
sinhx = coshx (ex+e-x)/2
coshx=sinhx (ex-e-x)/2
tanhx sech2x
sechx -tanhxsechx
cschx -cothxcschx
cothx -csch2x
arcsinhx=sinh-1x 1/√(x2+1)
arccoshx=cosh-1x 1/√(x2-1)
arctanhx=tanh-1x 1/(1-x2)
arccothx=coth-1x 1/(1-x2)
arccothx=coth-1x -1/(x∙√(1-x2))
arccothx=coth-1x -1/(|x|∙√(1+x2))


Example 1: Evaluate
Solution: Use the quotient rule


where u = x6 and v = sinh(x) +1


The derivative of a sum is the sum of the derivatives.


The derivative of the constant 1 is 0.


The derivative of sinh(x) is cosh(x)


The derivative of x6 is 6x6-1


Simplify the equation we get:


Example 2: Differentiate each of the following functions

a)      f(x) = 2x5cosh(x)

b)      h(t) = sinh(t)/(t+1)

Solution:
     a)    f´(x) = 10x4cosh(x) + 2x5sinh(x)

     b)  

 

Example  3:  Find the derivative of f(x) = sinh (x 2)

Solution :

Let u = x 2 and y = sinh u and use the chain rule to find the derivative of the given function f as follows.

f '(x) = (dy / du) (du / dx)

dy / du = cosh u, see formula above, and du / dx = 2 x

f '(x) = 2 x cosh u = 2 x cosh (x 2)

Substitute u = x 2 in f '(x) to obtain

f '(x) = 2 x cosh (x 2)




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