`D(1/((log)_x e))=\ ;\ (w h e n\ x=1/e)`

Comprehension 1 Let a^((log)_b x)=c w h e r e a , b , c & x are parameters. On the basis of above information, answer the following questions: If b=(log)_(sqrt(3))3, c=2(log)_bsqrt(b) a n d s intheta=a(w h e r e x >1) t h e n theta can be a.pi/4 b. (3pi)/2 c. pi/2 d. 0

The minimum value of x(log)_(e)x is equal to e (b) 1/e(c)-1/e (d) 2e( e )e

The minimum value of (x)/((log)_(e)x) is e(b)1/e(c)1(d) none of these

The differential coefficient of f((log)_e x) with respect to x , where f(x)=(log)_e x , is (a)x/((log)_e x) (b) 1/x(log)_e x (c)1/(x(log)_e x) (d) none of these

Integration of (1)/(1+((log)_(e)x)^(2)) with respect to (log)_(e)x is ((tan^(-1)((log)_(e)x))/(x)+C(b)tan^(-1)((log)_(e)x)+C(c)(tan^(-1)x)/(x)+C(d) none of these

If f(x)=cos^(-1){(1-(log_(e)x)^(2))/(1+(log_(e)x)^(2))} , then f'( e )

If x , y >0,(log)_y x+(log)_x y=(10)/3 a n d x y=144 , t h e n(x+y)/2=sqrt("N")" w h e r e N" is a natural number, find the value of Ndot

The primitive of the function f(x)=(1-(1)/(x^(2)))a^(x+(1)/(x)),a>0 is (a^(x+(1)/(x)))/((log)_(e)a) (b) (log_(e)adot a^(x+(1)/(x))(c)(a^(x+(1)/(x)))/(x)(log)_(e)a(d)x(a^(x+(1)/(x)))/((log)_(e)a)

If y=tan^(-1){((log)_(e)(e/x^(2)))/((log_(e)(ex^(2)))}+tan^(-1)((3+2(log)_(e)x)/(1-6(log)_(e)x)), then (d^(2)y)/(dx^(2))=(a)2(b)1(c)0(d)-1

If I_(n)=int_(1)^(e)(log x)^(n) d x, then I_(n)+nI_(n-1) equal to