arctan issues

[shizzy]

Having problems figuring out what to do with arctan here. It's a partial derivative with respect to x, but the problem is I just don't know how to manipulate arctan:

f(x,y) = y^-3/2 * tan^-1(x/y)
differentiate dz/dx.

I've been working on this for about 2 hours and give up now!! I have tried all kinds of stuff but just don't know if what I'm doing is okay or not. Basically I don't know how to even start this problem b/c I can't figure out what to do with arctan....PLEASE help me!!

 

[tkhunny]

Unfortunately, you haven't defined 'z', so my first guess is dz/dx = 0.

y is a function of x? or not?

 

[soroban]

Hello, shizzy!

Having problems figuring out what to do with arctan here.
It's a partial derivative with respect to x,
but the problem is I just don't know how to manipulate arctan:

z .= .y^-3/2tan^-1(x/y)

Find dz/dx.

You know the derivative of: .y .= .arctan u

. . . . . dy . . . . . . 1 . . . .du
It is: . --- . = . --------- . ----
. . . . . dx . . . . 1 + u² . .dx

. . . . . . . . . . . . . . . . . . . . . . . . . . 1
In your problem: . u .= .(x/y) .= .--- x
. . . . . . . . . . . . . . . . . . . . . . . . . . y

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 . . . . . 1
The derivative of: . arctan(x/y) . = . ------------- . --- . . . . . . . and so on . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 + (x/y)² . . y

 

[shizzy]

sorry about that guys. The website went down and I ended up figuring it out about 1 hour later. Then I spent four more hours proving that the derivative of

arctan(x/a) = a/(x² + a²).

Fun stuff.