Question

Question #6aa72

Answer 1

`[(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.