Question

求下列函數的導數：

(1)y=x⁴+3x²-6;

(2)y=6+4+2x;

(3)y=x(2x-1)(3x+2);

(4)y=xsinx+cosx.

Answer 1

分析：這些函數都是由基本初等函數經過四則運算得到的簡單函數，求導時可直接利用求導法則和導數公式進行求導.

解：(1)y′=(x⁴+3x²-6)′

=(x⁴)′+(3x²)′-(6)′

=4x³+6x.

(2)y′=(6x+4x+2x)′

=(6x)′+(4x)′+(2x)′

=6×x+4×x+2

=21x+10x+2.

(3)y′=［x(2x-1)(3x+2)］′

=［x(2x-1)］′(3x+2)+x(2x-1)(3x+2)′

=x′(2x-1)(3x+2)+x(2x-1)′(3x+2)+x(2x-1)(3x+2)′

=(2x-1)(3x+2)+2x(3x+2)+3x(2x-1)

=18x²+2x-2.

(4)y′=(xsinx+cosx)′

=(xsinx)′+(cosx)′

=x′·sinx+(sinx)′·x+(cosx)′

=sinx+xcosx-sinx

=xcosx.