Finding a function for rate of change

[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

 

[HallsofIvy]

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 $df/dx= (df/du)(du/dx)= (1/(1+ u^2))(a sec^(x)= (a sec^2(x))/(1+a^2tan^2(x)$.

 

[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.