Differentiate the function q(x)= xtan^-1 (x) - (1/2)ln(1+x^2)

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`q(x)= xtan^(-1) (x) - (1/2)ln(1+x^2)`

Using function of function or chain rule you can solve this.

`(dtan(-1)x)/dx = 1/(1+x^2)`

`(dq(x))/dx = (d(xtan^(-1)x))/dx-1/2(d(log(1+x^2)))/dx`

`(dq(x))/dx = x*1/(1+x^2)+tan^(-1) (x)*1-1/2*1/(1+x^2)*2x`

`(dq(x))/dx = x/(1+x^2)+tan^(-1) (x)-x/(1+x^2)`

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