How do you find the derivative of exponential function #f(x)= 2 / (e^x + e^-x)^3#?

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Answer 1 · 2018-04-10T12:02:39

#d/dx (2/(e^x+e^-x)^3) = -(6 (e^x-e^-x))/(e^x+e^-x)^4=-(3sinh x)/(4cosh^4x)#

Using the chain rule:

#d/dx (2/(e^x+e^-x)^3) = 2 d/dx((e^x+e^-x)^-3)#

#d/dx (2/(e^x+e^-x)^3) = 2(-3)(e^x+e^-x)^-4 d/dx (e^x+e^-x)#

#d/dx (2/(e^x+e^-x)^3) = -6(e^x+e^-x)^-4 (e^x-e^-x)#

#d/dx (2/(e^x+e^-x)^3) = -(6 (e^x-e^-x))/(e^x+e^-x)^4#

Note also that:

#2/(e^x+e^-x)^3 = 1/4 1/((e^x+e^-x)/2)^3#

#2/(e^x+e^-x)^3 = 1/(4cosh^3x)#

#d/dx (2/(e^x+e^-x)^3) =-(3sinh x)/(4cosh^4x)#