(1) If you can’t remember the derivative of arcsec, here’s how you can do this:
sec(y)=√x, sec²(y)=x, 1+tan²(y)≡sec²y, tan(y)=√(sec²(y)-1)=√(x-1);
sec(y)tan(y)y'=1/(2√x),
(√x)(√(x-1))y'=1/(2√x),
y'=dy/dx=1/(2x√(x-1)), which can also be written ½√(x-1)/(x²-x).
(2) Let y=ux, dy/dx=y'=u+u'x;
y/x=arctan(4x), u=arctan(4x), tan(u)=4x, sec²(u)u'=4, (1+tan²(u))u'=4, (1+16x²)u'=4, u'=4/(1+16x²);
y'=u+u'x=arctan(4x)+4x/(1+16x²).
(3) y=arccsc(x³), csc(y)=x³, sin(y)=1/x³, cos(y)=√(1-1/x⁶)=√(x⁶-1)/x³;
y'cos(y)=-3/x⁴, y'√(x⁶-1)/x³=-3/x⁴, y'√(x⁶-1)=-3/x, y'=-3/(x√(x⁶-1)) which can be written -3√(x⁶-1)/(x⁷-x).
(4) y=arccot(1-2x), cot(y)=1-2x, tan(y)=1/(1-2x), tan²(y)=1/(1-2x)², sec²(y)=1+1/(1-2x)²=(2-4x+4x²)/(1-2x)²=2(1-2x+2x²)/(1-2x)²;
y'sec²(y)=2/(1-2x)², 2(1-2x+2x²)/(1-2x)²y'=2/(1-2x)², y'=1/(1-2x+2x²).