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How would you do Q10cii? Thanks

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It's probably easier to use the results from the previous two questions rather than starting from exponential definitions. Differentiating (10ci) gives h''(x) in terms of g(x) and g'(x), which you can simplify using the chain rule plus an equation you should already have for g'(x). From there you should be able to finish it off using the definitions.
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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.

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