Question #6aa72

1 Answer

#[(ln(ln x))/(1+x^2)+(tan^(-1)x)/(xln x)](ln x)^(tan^(-1)x)#

Explanation:

By taking the natural log of both sides,

#ln(y)=ln(ln x)^(tan^(-1)x)#

By the log property: #ln x^r=r ln x#,

#Rightarrow ln(y)=tan^(-1)x cdot ln(ln x)#

By differentiating using Product Rule,

#Rightarrow (y')/y=(tan^(-1)x)'cdot ln(lnx)+tan^(-1)x cdot(ln(ln x))'#

By #(tan^(-1)x)'=1/(1+x^2)# & #[ln(g(x))]'=(g'(x))/(g(x))#,

#Rightarrow (y')/y=1/(1+x^2) cdot ln(ln x)+tan^(-1)x cdot (1/x)/(ln x)#

By multiplying both sides by #y#,

#y'=[(ln(ln x))/(1+x^2)+(tan^(-1)x)/(xln x)](ln x)^(tan^(-1)x)#

I hope that this was clear.