Calculating average every from partition function

[planesinspace]

1. I can't seem to get the same answer my textbook does, basically I need to calculate E (average energy) from the Partition function (Z) which is defined as:

E=(-1/Z)*(dZ/dBeta)

Where Z=(1/1-exp(-Beta*h*f))
(where h and f are constants and beta=1/kT for simplicity)


So for my differentiation I get:
dZ/dBeta= -h*f*exp(-Beta*h*f)) / (1-2exp(-Beta*h*f))+exp(-2*Beta*h*f))

Which when multiplied by 1/Z gives:
-h*f + h*f*exp(-Beta*h*f) / (1-exp(-Beta*h*f) + exp(-2*Beta*h*f)

When the answer is apparently:

(h*f) /(exp(Beta*h*f) -1 )


Any help greatly appreciated!

 

[kuruman]

Note that

$$\frac{dZ}{d \beta}=-hfe^{-\beta h f}Z^2$$

Does this help?