# How can you prove that d/dx(cothx) = -csch^2x using the definition cothx=coshx/sinhx?

Keeping in mind that the hyperbolic derivatives are ##d/dx (sinh(x))=cosh(x)## and ##d/dx (cosh(x))=sinh(x)## and that the fundamental hyperbolic identity is ##cosh(x)^2-sinh(x)^2=1##, and by using the , we find:

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How can you prove that d/dx(cothx) = -csch^2x using the definition cothx=coshx/sinhx?
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##d/dx(coth(x))=d/dx(cosh(x)/sinh(x))=##
##(d/dx(cosh(x))sinh(x)-cosh(x)d/dx(sinh(x)))/(sinh(x)^2) =##
##(sinh(x)^2-cosh(x)^2)/(sinh(x)^2) = -1/(sinh(x)^2)=-csch(x)^2##

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