f(x)= x/sqrtsinx
by Quotient Rule:
f'(x)= ( x' sqrtsinx - x (sqrtsinx)' ) /(sqrtsinx)^2 = ( 1* sqrtsinx - x (sqrtsinx)' ) /(sinx)
the oddest bit:
(sqrtsinx)' = 1/2 1/(sqrtsinx) cos x
so we have
f'(x)= ( sqrtsinx - x(1/2) 1/(sqrt sinx) cos x) /(sinx)
= ( sinx - 1/2 x cos x) /(sinx)^{3/2}
f'(pi/8) = ( sin(pi/8) - 1/2 (pi/8) cos (pi/8)) /(sin(pi/8))^{3/2} = 0.85024
that is the slope, s, of the tangent. slope of the normal is -1/s = -1.178
f(pi/8) = 0.634
y = mx + c \implies 0.634 = -1.178(pi/8) + c \implies c = 1.097
So the line is: y = -1.178x + 1.097
![Desmos]()
