Finding a function for rate of change

1. Jul 12, 2013

PennyPuzzleBox

Hello!

I was wondering if somebody could help me develop an answer to the question below.

I would like to calculate a rate of change of the following function:

θ = arctan (a * (tan β)) where "a" is a constant.

By the way, the shape of the curve -- angle alpha as a function of angle Beta --resembles an arm of a parabola. Angle Beta increases with angle alpha, but less so with increasing values.

I would like to derive a function that would describe the instantaneous rate of change at each point of that curve. It would be great if the function would say: hey, this is an arm of a parabola! :)

Unfortunately I took math a very long time ago... I believe I would
need to find the derivative of the function? Could you kindly help me out with this?
thnx

Penny

2. Jul 12, 2013

HallsofIvy

Staff Emeritus
The whole point of the "derivative" is that it is the rate of change of a function. The derivative of arctan(x), with respect to x, is $1/(1+ x^2)$. The derivative of tan(x) is $sec^2(x)$.

So, by the "chain rule", with f= arctan(u) ,u= a tan(x), $df/du= 1/(1+ u^2)$ and $du/dx= a sec^2(x)$, so $df/dx= (df/du)(du/dx)= (1/(1+ u^2))(a sec^(x)= (a sec^2(x))/(1+a^2tan^2(x)$.

3. Jul 13, 2013

nickbob00

^ What HallsOfIvy said

If you're looking for an easy way to find derivatives and do other mathsy things, look up wolfram alpha. It's a website that is quite excellent.

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook