#d/dx(tan^-1(1.5/x)-tan^-1(0.5))=d/dx(tan^-1(1.5x^-1)-tan^-1(0.5))#
When differentiating, #- tan^-1(0.5)# goes away as it's just a constant value. The derivative of any constant is #0.#
#d/dx(tan^-1(1.5x^-1)-tan^-1(0.5))=(1/(1+(1.5x^-1)^2))*d/dx(1.5x^-1)#
Using the Chain Rule along with the fact that
#d/dxtan^-1(x)=1/(1+x^2)#. Here, we have #1.5x^-1# as the argument of the arctangent, so instead of #x^2#, we will have #(1.5x^-1)^2# in the denominator.
#(1/(1+(1.5x^-1)^2))*d/dx(1.5x^-1)=1/(1+2.25x^-2)*-1.5x^-2#
Simplify:
#-1.5/(x^2(1+2.25/x^2))=-1.5/(x^2+(2.25cancelx^2)/cancelx^2)=-1.5/(x^2+2.25)#
#1.5=3/2#
#2.25=9/4#
Getting rid of the decimals:
#-1.5/(x^2+2.25)=-(3/2)/(x^2+9/4)=-3/(2(x^2+9/4))=-3/(2x^2+9/2)=-3/((4x^2+9)/2)=-3*(2/(4x^2+9))= -6/(4x^2+9)#
