QUESTION: find the value of dy/dx at x=0.5 for the curve y=3cos^-1 (x) .
y = 3cos^-1 (x)
y/3 = cos^-1 (x) .
cos(y/3) = x
differentiating both sides wrt y,
(-1/3)sin(y/3) = dx/dy
sqrt(1- cos^2(y/3)) = -3.dx/dy
sqrt(1-x^2) = -3.dx/dy
dy/dx = -3/{sqrt(1-x^2)}
at x = 0.5,
dy/dx = -3/{sqrt(1-1/4)} = -3/{sqrt(3)/2}
dy/dx = -6/sqrt(3) = -2*sqrt(3)
Answer: -2*sqrt(3)