arctan(y/x)=( 1/2)ln(x^2+y^2)

[ 1/( 1+ (y/x)^2) ]。 (x.dy/dx-y)/x^2 =( 1/2)[ 1/(x^2+y^2)][2x+2y dy/dx]

[ 1/(x^2+ y^2]]。 (x.dy/dx-y)=[ 1/(x^2+y^2]][x+y.dy/dx]

x.dy/dx-y = x+y.dy/dx

(x-y) dy/dx = x+y

dy/dx = (x+y)/(x-y)

(2)

siny =ln(x+y)

cosy.dy/dx =[ 1/(x+y)]。 ( 1+dy/dx)

(x+y)cosy.dy/dx = 1+dy/dx

[(x+y)cosy- 1]。 dy/dx = 1

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

( 1)

y=x+x^x+ x^(x^x)

let

p=x^x

lnp=xlnx

( 1/p)dp/dx = 1+ lnx

dp/dx =( 1+lnx)x^x

q=x^(x^x)

lnq = (x^x)lnx

( 1/q)dq/dx = x^(x- 1)+lnx。 ( 1+lnx)x^x

dq/dx =[x^(x- 1) + lnx。 ( 1+lnx)x^x].x^(x^x)

y=x+x^x+ x^(x^x)

dy/dx = 1 +( 1+lnx)x^x +[x^(x- 1)+lnx。 ( 1+lnx)x^x].x^(x^x)