If `y=10^(x)` and `z=100^(x/2)` then find `d/dx(y^(2)/z)`.

If x=4z^(2)+5 and y=6z^(2)+7z+3 , then find (d^(2)y)/(dx^(2))

If y=sec^(-1)((x^(2)+1)/(x^(2)-1)) , then find (dy)/(dx) . Here f^(-1)(x) expression is of the form (x^(2)+a^(2))/(x^(2)-a^(2)) , so we substitute x=tan theta and then use suitable trigonometrical formula to write it in simplest form and then differentiate

If e^(y//x)=(x)/(a+bx) then show that x^(3)(d)/(dx)((dy)/(dx))=(x(dy)/(dx)-y)^(2)

If y= tan^-1 x then prove that (1+x^2) y_(2) + 2xy_(1) =0. Differentiate sec^-1(1/(2x^2 -1)) with respect to sqrt 1-x^2 .

Verify that the function y= e^(-3x) is a solution of the differential equation (d^(2)y)/(dx^2)+(dy)/(dx)-6y=0