Calculator Derivative Error at 11: What's Causing the Drastic Difference?

[lpbug]

Hi everyone, I've discovered quite a strange function in my textbook. I compared the calculator's calculated derivatives (Math->nDriv) with my calculus derivative and found that the two are almost 2000 units off! Up until now I've been trying to figure out the problem, but to no avail, I cannot find the solution. Here's the function:

f(x)=Cos2(tanx)

using derivative shortcuts, I came up with

f'(x)=-2cos(tanx)sin(tanx)sec2x

as my derivative function

plugging in 11, I came up with -23921.7403... as the derivative at 11

however, when i compare this with the calculator's nDriv answer, it is drastically different:
nDriv(f(x), x, 11)= -342.6

that's difference in the 2000 units... i mean, i compared the rest of the values at various other points and the most off they have been is around .001; only at 11 does this great gap occur.

Does anyone have any idea what the problem is? Any tips and help will be appreciated. If you have a ti-89 or another version of the calculator, please feel free to try it on your calculator and post if you have the same result as me.

Thanks so much,
Alex

 

[lanedance]

i assume you mean
$$f(x)=cos^2(tan(x))$$

differentiating
$$f'(x)=-2cos(tan(x))sin(tan(x)) \frac{1}{cos^2(x)}$$

2 things to note
cos(11) = 0.00442
which is a small number, as we are close to zero the value of the function will be diverging quite quickly. I would trust the analytic solution...

the other thing to check is whether you are in degrees or radians

 

[lpbug]

that's really weird because I've never seen this happen with any other function, is there an example that you can give where the result will be replicated?

 

[HallsofIvy]

Any time you are dividing by a very small number (cos(11)= 0.0044257), round off error becomes important.