高数求指导1.已知g(x)=1/x^2且复合函数f(g(x))对x的导数为-1/2x,那么f'(1/2)=A.1B.-√2/4C.1/2D22.设函数y=y(x)由方程2^(xy)=x+y确定,则dy|x=0=3.由方程e^y+xy-e=0确定的y是x的函数,求y'|x=0

Question

高数求指导
1.已知g(x)=1/x^2 且复合函数f(g(x))对x的导数为-1/2x,那么f'(1/2)=
A.1 B.-√2/4 C.1/2 D 2
2.设函数y=y(x)由方程2^(xy)=x+y 确定,则 dy | x=0 =___
3.由方程e^y+xy-e=0确定的y是x的函数,求 y' | x=0

Answer 1

1.已知g(x)=1/x^2 且复合函数f(g(x))对x的导数为-1/2x,那么f'(1/2)=
A.1 B.-√2/4 C.1/2 D 2
d(f(g(x))/dx=-x/2
f(g(x))=-x^2/4+C
f(1/x^2)=-x^2/4+C
1/x^2=t x^2=1/t
f(t)=-1/(4t)+C
f(x)=-1/(4x)+C
f'(x)=1/(4x^2)
f'(1/2)=1/(4*1/4)=1
2.设函数y=y(x)由方程2^(xy)=x+y 确定,则 dy | x=0 =___
x=0 2^0=0+y y=1
ln2^(xy)=ln(x+y)
xyln2=ln(x+y)
yln2+xy'ln2=y'/(x+y)
y'(xln2-1/(x+y))=yln2
y'=yln2/[xln2-1/(x+y)]
dy=yln2/[xln2-1/(x+y)]dx
dy/dx |x=0 =ln2/[0-1]dx=-ln2
3.由方程e^y+xy-e=0确定的y是x的函数,求 y' | x=0
x=0 e^y+0-e=0 e^y=e y=1
e^y * y'+y+xy'=0
y'=-y/(e^y+x)
y'(0)=-1/(e+0)=-1/e