Show that , `d/(dx)[x/2sqrt(a^2-x^2)+a^2/2 sin^(-1)x/a]=sqrt(a^2-x^2)`

Show that, (d)/(dx) [(x)/(2) sqrt(a^(2)-x^(2)) +(a^(2))/(2) "sin"^(-1) (x)/(a)]=sqrt(a^(2)-x^(2))

Differentiate tan^(-1)(x/sqrt(1-x^2)) with respect to sin^(-1)((2x)*sqrt(1-x^2)),

d/(dx)[cos^(-1)(xsqrt(x)-sqrt((1-x)(1-x^2)))]= 1/(sqrt(1-x^2))-1/(2sqrt(x-x^2)) (-1)/(sqrt(1-x^2))-1/(2sqrt(x-x^2)) 1/(sqrt(1-x^2))+1/(2sqrt(x-x^2)) 1/(sqrt(1-x^2)) 0 b. 1//4 c. -1//4 d. none of these

Evaluate, d/(dx)[sin^2cot^(- 1)sqrt((1-x)/(1+x))] is

State whether the following statement is true or false : (d)/(dx)(cos^(-1)x-sin^(-1)x)=(2)/(sqrt(1-x^(2))) , when |x|lt1 .

Differentiate tan^(-1)(x/(1+sqrt((1-x^2)))) +{2tan^(-1)sqrt(((1-x)/(1+x)))} w.r.t. x

For each of the following differential equations verify that the accompanying functions a solution. Differential Function (a) x(dy)/(dx)=y, y=a x (b) x+y(dy)/(dx)=0, y=+-sqrt(a^2-x^2) (c) x(dy)/(dx)y=y^2, y=a/(x+a) (d) x^3(d^2y)/(dx^2)=1, y=a x+b+1/(2x) (e) y=((dy)/(dx))^2, y=1/4(x+-a)^2

The function y=f(x) is the solution of the differential equation (dy)/(dx)+(x y)/(x^2-1)=(x^4+2x)/(sqrt(1-x^2)) in (-1,1) satisfying f(0)=0. Then int_((sqrt(3))/2)^((sqrt(3))/2)f(x)dx is

The integrating factor of the linear differential equation (1-x^(2))^(2)(dy)/(dx) + sqrt(1-x^(2))y=x+sqrt(1-x^(2)) is -

The function y=f(x) is the solution of the differential equation (dy)/(dx)+(xy)/(x^(2)-1)=(x^(4)+2x)/(sqrt(1-x^(2))) in (-1, 1) satisfying f(0)=0 . Then underset((-sqrt(3))/(2))overset((sqrt(3))/(2))intf(x)dx is