Question :

(a) let y = cos x, find d^999y/dx^999

(b) let y = x^x^x^x^x^x...., find dy/dy

(c) let y =sqrt(x+sqrt(x+sqrt(x+sqrt(x+......), find dy/dx

my attempt :

(a) y= cos x

y' = -sin x

y'' = -cos x

y'''= sin x

y^(IV) =cos x = y

thus, y^(4n) = y

then 999/4 = 249 bal 3

999 = (249x4)+3

thus, d^999y/dx^999 = y^(999)

= y^(249x4)+3

= y^(996).y^(3)

= cos x sin x

(b) y = x^x^x^x^x^x.... can be expressed as y=x^y

taking log both side log y = y log x....(*)

diff * w.r.t. x :

1/y(dy/dx) = y(1/x) +log x (dy/dx)

(1/y-log x )(dy/dx) =y/x

[(1-y log y)/y] dy/dx = y/x

dy/dx =(y^2)/x(1-y log x)

(c) y =sqrt(x+sqrt(x+sqrt(x+sqrt(x+......) can be expressed as y = sqrt(x+y)

squaring both side

y^2 = x+y ....(*)

diff (*) w.r.t. x :

2y(dy/dx) = 1 +dy/dx

(2y-1) (dy/dx) = 1

dy/dx = 1/(2y-1)

IS MY ANSWER CORRECT, i'm afraid my answer for (a) is wrong... can anyone clariffy ot for me??? THANX...(Speechless)