Did I Apply the Taylor Series Correctly to the Restoring Force?

[NeedPhysHelp8]

Homework Statement
An atom of mass m is bonded to surface immobile body by electromagnetic forces. The force binding the atom to the surface has the expression:

$$F= exp\ (a\cos z + b\sin z) + d\tan(z)$$

where a,b, and d are constants and z is upwards. The equilibrium point is defined to be z=0. The system is subject to Earth's gravity

For small oscillations, give an approximate expression for the binding force on the atom.

The attempt at a solution
Ok so in class we were taught to do the Taylor Series expansion on F(x) and keep the linear term so you can get something that looks like F=-kx . So I got this after doing Taylor Series expansion:

where $$F' = (-a\sin z + b\cos z) exp\ (a\cos z + b\sin z) + d\sec^2 (z)$$ then evaluate at z=0 then final result is :

$$F(z)= (b exp(a) + d )z$$

Since $$F(0)$$ is always 0 at the equilibrium I took that out of Taylor series and ignored higher order terms.

Now I'm not sure if I'm going about this the right way? Can someone please tell me if I did this right or not? Much appreciated I love this forum!

 

[diazona]

You've got the right idea, but you seem to have computed F' incorrectly. Take another look at that calculation.

 

[NeedPhysHelp8]

How is F' wrong?? :S please explain I just used the chain rule

 

[diazona]

It's wrong for the F that was in your post when I wrote mine. But after your edit, everything looks OK.

 

[NeedPhysHelp8]

Thanks