Question

求下列函数的偏导数：（1）z=√ln(xy) （2）S=u+v/u-v （3）u=x^(y/z) （4）z=(1+xy)^y

Answer 1

(1)偏z/偏x=1/[2x√ln(xy)]偏z/偏y=1/[2y√ln(xy)](2)偏s/偏u=[(u+v)'(u-v)-(u+v)(u-v)']/(u-v)^2=(u-v-u-v)/(u-v)^2=-2v/(u-v)^2偏s/偏v=[(u+v)'(u-v)-(u+v)(u-v)']/(u-v)^2=(u-v+u+v)/(u-v)^2=2u/(u-v)^2(3)偏u/偏x=(y/z)*x(y/z-1)偏u/偏y=x^(y/z)*(y/z)'*lnx=(1/z)x^(y/z)*lnx偏u/偏z=x^(y/z)*(y/z)'*lnx=(-y/z^2)x^(y/z)*lnx(4)偏z/偏x=y(1+xy)^(y-1)(1+xy)'=y^2*(1+xy)^(y-1)偏z/偏y={e^[yln(1+xy)]}'=(1+xy)^y*[ln(1+xy)+xy/(1+xy)]