Question

求下列各函數的導數:
(1)y=(x+1)(x+2)(x+3).
(2)y=+.
(3)y=e^-xsin2x.

Answer 1

(1) y'=3x²+12x+11 (2) y'=.  (3) y' =e^-x(2cos2x-sin2x).

(1)方法一:y=(x²+3x+2)(x+3)=x³+6x²+11x+6,∴y'=3x²+12x+11.
方法二:y'=[(x+1)(x+2)]'(x+3)+(x+1)(x+2)·(x+3)'
=[(x+1)'(x+2)+(x+1)(x+2)'](x+3)+(x+1)·(x+2)
=(x+2+x+1)(x+3)+(x+1)(x+2)
=(2x+3)(x+3)+(x+1)(x+2)
=3x²+12x+11.
(2)∵y=+=,
∴y'=()'==.
(3)y'=(-e^-x)sin2x+e^-x(cos2x)×2
=e^-x(2cos2x-sin2x).