What is the derivative of # x ln(arctan x)#?

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Answer 1 · 2018-03-17T23:09:40

The derivative is #=ln(arctan(x))+x/((1+x^2))*1/(arctan(x))#

Let #u=arctanx#

Then,

#tanu=x#

#sec^2u=1+x^2#

Differentiating wrt #x#

#(du/dx)*sec^2u=1#

#(du)/dx=1/sec^2u=1/(1+x^2)#

Therefore,

#y=xln(arctanx)#

#y=xln(u)#

Differentiating wtt #x# by pplying the product rule

#dy/dx=1*lnu+x*(du)/dx*1/u#

#=ln(arctan(x))+x/((1+x^2))*1/(arctan(x))#

Answer 2 · 2018-03-17T23:12:33

#d/dxxlnarctanx=lnarctanx+x/((1+x^2)arctanx)#

To find #d/dxxlnarctanx#, we use the product rule

#d/dxuv=v(du)/dx+u(dv)/dx#

#d/dxlnarctanx=1/((1+x^2)arctanx)#

So

#d/dxxlnarctanx=lnarctanx+x/((1+x^2)arctanx)#