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:

##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##