Question #52b72

1 Answer
Feb 14, 2018

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)#