Find the derivative of y=10csc(3x/2) at x=2pi/6? Solution We\'ll recall the identity that gives the cosecant function: csc x = 1/sin x y = 10csc(3x/2) = 10/sin(3x/2) We\'ll calculate the 1st derivative of the function using the quotient rule: dy/dx = [(10)\'*sin(3x/2) - 10*[sin(3x/2)]\'/[sin(3x/2)]^2 dy/dx = - 15*[cos(3x/2)]/[sin(3x/2)]^2 Now, we\'ll calculate the value of the first derivative at x = 2 [pi] /6 dy/dx = - 15*[cos(6 [pi] /12)]/[sin(6 [pi] /12)]^2 dy/dx = -15*cos( [pi] /2)/(sin [pi] /2)^2 But cos [pi] /2 = 0, therefore dy/dx = 0 at x = 2 [pi] /6. Therefore the function has an extreme at x = 2 [pi] /6. The requested value of the 1st derivative, at x = 2 [pi] /6 is dy/dx = 0..