求函数的导数

（1）求x^2/a^2-y^2/b^2=1上点（x0,y0）处切线方程． 对曲线方程的两边取导数，得2x/a^2+2yy'/b^2=0--->y'=-(b^2*x)/(a^2*y)--->y'(x0,y0)=-(b^2*x0)/(b^2*y0)由直线的点斜式，切线方程是y-y0=[-(a^2*x0)/(b^2*y0)](x-x0)--->a^2*y0y-a^2*y0^2=-b^2*x0x+b^2*x0^2--->b^2*x0x+a^2*y0y=b0^2*x0^2+b^2*y0^2--->x0x/a^2+y0y/b^2=x0^2/a^2+y0^2/b^2因为点(x0,y0)在曲线上所以x0^2/a^2+y0^2/b^2=1因此切线方程是x0x/a^2+y0y/b^2=1（2）求曲线x=3t/(1+t^2),y=3t^2/(1+t^2)上切线垂直x轴和切线平行x 轴的点．x'=[3(1+t^2)-3t*2t]/(1+t^2)^2=3(1-t^2)/(1+t^2)^2y'=3[2t(1+t^2)-t^2*2t]/(1+t^2)^2=6t/(1+t^2)^2--->y'(x)=y'/x'=2t/(1-t^2)切线平行于x轴时，y'=0(y'(x)=0)--->t=0,此时x=0,y=3,切点是(0,3)切线垂直于y轴时，x'=0(y'(x)不存在)--->t=+'-1,此时x=3/2,,y=3/2或x=-3/2,y=3/2,切点是(+'-3/2,3/2). 收起

(1)对隐函数求导,可得(x0,y0)处斜率k=-b^2x0/a^2y0,故切线为y-y0=-b^2x0/a^2y0*(x-x0) ==>xx0/a^2+yy0/b^2=1。(2)k=y'(t)/x'(t)=2t/(1-t^2),t=0时,即(0,0)处切线平行于X轴;1-t^2=0 ==>t=1或-1时,切线平行于Y轴,此时以t值代入参数方程可得切点为(3/2,3/2)或(-3/2,3/2) 收起