Question

Question #52b72

Answer 1

By using the common derivative arccot formula and the chain rule, we get `alpha/(alpha^2+x^2)`

Explanation:

For the given problem:

`y=`arccot`(alpha/x)`

We must apply a common derivative formula of:
`d/dx(`arccot`(u)`)`=-1/(u^2+1)`

and the chain rule (this is only because our "u" is something other than just x)

1. in our problem we assign the following for ease:
   `f=(`arccot`(u))`
   `u=alpha/x`

2. our derivative will be:
   `=d/(du)(`arccot`(u)`)`*d/dx(alpha/x)`

3. We can then calculate them separately and simplify:
   `d/(du)(`arccot`(u)`) = `color(red)(-1/((alpha/x)+1))`
   `d/(dx)(alpha/x)=alpha*d/(dx)1/x=alpha*d/(dx)x^-1=color(blue)(-alpha/x^2)`

4. Multiply together and simplify
   `color(red)(-1/((alpha/x)+1))` `*` `color(blue)(-alpha/x^2)`

For a final answer of:
`=alpha/(alpha^2+x^2)`