If f(x)`=tan^(-1)((x)/(1+20x^(2)))`, show that,
`f'(x)=(5)/(1+25x^(2))-(4)/(1+16x^(2))`

If f(x)=tan^(-1)((3x-x^(3))/(1-3x^(2))) and phi(x)=cos^(-1)((1-x^(2))/(1+x^(2))) , then the value of lim_(x to a) (f(x)-f(a))/(phi(x)-phi(a))(0 lt a lt (1)/(2)) is -

If f(x)=((a+x)/(1+x))^(a+1+2x) , show that f'(0) =a^(a+1)[2log a+(1-a^(2))/(a)] .

If f(x)=log_e.(1+x)/(1-x) , show that f((2x)/(1+x^2))=2f(x) .

If f(x) =tan ^(-1)x-(1)/(2)log x . Then -

If f(x)=(1)/(x^2) , show that f(x+h)-f(x-h)=-(4xh)/((x^2-h^2)^2) .

If 2f(x-1)-f((1-x)/x)=x , then prove that 3f(x)=2(x+1)+1/(x+1) .

If f(x)=(4^x)/(4^x+2), then show that f(x)+f(1-x) =1.

If f(x)=x/(2^(x)-1) , then show that f(-1) = 2.

If y="tan"^(-1)(4x)/(1+5x^(2))+"tan"^(-1)(2+3x)/(3-2x) , prove that, (dy)/(dx)=(5)/(1+25x^(2))

If f(x)=log[(1+x)/(1-x)], then prove that f[(2x)/(1+x^2)]=2f(x)dot