A) y=(3+6x)/(√(3-4x+5x²)
6√(3-4x+5x²)-(3+6x)(-4+10x)
-----------------
2√(3-4x+5x²) 6(3-4x+5x²)-(3+6x)(-2+5x)
y' = -------------------------------------------- = -----------------------------------------=
(3-4x+5x²) (3-4x+5x²) √(3-4x+5x²)
30x²-24x+18 +6+12x-15x-30x² -27x +24
= ----------------------------------------- = ---------------------------------------
(3-4x+5x²) √(3-4x+5x²) (3-4x+5x²) √(3-4x+5x²)
б) у=sinx-xcosx y'= cosx-(cosx-x·sinx)
в) у=(x^m)㏑x
y'=mx^(m-1)㏑x +(x^m)/x= mx^(m-1)㏑x +x^(m-1)=x^(m-1)(m㏑x +1)
г) y=x^(-tgx)
ln y = -tgx ·lnx
y'/y = -[(1/cos²x)lnx+(tgx)/x] ⇒ y' = -y[(1/cos²x)lnx+(tgx)/x]
y' = -x^(-tgx)·[(1/cos²x)lnx+(tgx)/x]
д) y/x= arctg(x/y)
(y'x -y)/x²={1/(1+(x/y)²)}·{(y-y'x)/y²}
y'·(1/x+x/(x²+y²))=y/(x²+y²)-y/x²
y'=y(1/(x²+y²)-1/x²)/(1/x+x/(x²+y²))
y'=-y³/(x(x²+y²-1))