It's worth knowing that these are the hyperbolic functions, sinh x, cosh x, tanh x. I don't know why they've chosen to hide this from you. The analogy with sin, cos, tan may help you find ideas about how to do things. So I'm going to write sinh, cosh and tanh everywhere for f, g, h.

We have sinh' = cosh, cosh' = sinh, tanh x = (sinh x)/(cosh x).

For (c)(i), using the quotient rule you find tanh'(x) = \[sinh'(x).cosh x - cosh'(x).sinh x\]/cosh\^2 x = (cosh\^2 x - sinh\^2 x)/cosh\^2 x. At that point, proving that tanh'(x) = 1/(cosh\^2 x) reduces to proving the identity cosh\^2 x - sinh\^2 x = 1, which I presume you've done.

Then for (c)(ii), tanh''(x) = -2 cosh'(x)/(cosh\^3 x) = -2(sinh x)/cosh\^3 x. So the problem reduces to proving the identity sinh 2x = 2 sinh x cosh x, which can be done by multiplying together the expressions for sinh x and cosh x.