微積分(微分學)

Question

若ln(x平方+y平方)+2arctan(x/y)=0,dy/dx=?
Ans: x+y/x-y
請會的人幫忙 ..thank you

Answer 1

ln(x²+y²)+2tan-1(x/y) = 0隱微分法(1/(x²+y²))(2x+2ydy/dx)+2(1/(1+(x/y)²)(y-xdy/dx)/y²=0剩下只是化簡(x+ydy/dx)/(x²+y²)+(y²/(x²+y²))(y-xdy/dx)/y²=0(x+ydy/dx)/(x²+y²)+(y-xdy/dx)/(x²+y²)=0x+ydy/dx+y-xdy/dx=0x+y-(x-y)dy/dx=0dy/dx=(x+y)/(x-y)　註： d(xn)/dx=nxn-1　　　d(lnx)/dx=1/x　　 d(tan-1x)/dx=1/(1+x²)　　 d(f(x)/g(x))/dx=(g(x)(df(x)/dx)-f(x)(dg(x)/dx))/(g(x))²　　 dy/dx=(dy/du)(du/dx)