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求微分這三題~感謝

Nov 9th 2014, 04:21

1.
   y=e^ { (x^2)/2 } + e^{ √(x^2)  }
     =e^ { (x^2)/2 } +e^lxl
   x>0， y= e^ { (x^2)/2 } +e^x
dy/dx =x e^ { (x^2)/2 } + e^x　
    x            dy/dx=x e^ { (x^2)/2 } - e^(-x)
    x=0，y= e^ { (x^2)/2 } +e^lxl= e^0 +e^0 =2
             dy/dx = lim [ e^ { (h^2)/2 } +e^lhl -2 ]/h , (h→0)
                h→0+ ， dy/dx = lim (e^ { (h^2)/2 } +e^h -2 )/h
0/0 type，use L'Hospital's rule
                        dy/dx=lim [he^{ (h^2)/2 } +e^h]/1 = 1
                h→0 - ， dy/dx = lim (e^ { (h^2)/2 } +e^(-h) -2 )/h
0/0 type，use L'Hospital's rule
                        dy/dx=lim [he^{ (h^2)/2 } -e^(-h)]/1 = -1
               左極限≠右極限，dy/dx =lim [ e^ { (h^2)/2 } +e^lhl -2 ]/h
   於x=0，不存在
Ans : dy/dx= x e^ { (x^2)/2 } + e^x　 ,for x>0
   = x e^ { (x^2)/2 } - e^(-x) ,for x    不存在， for x=0

2.
xy+sin (xy) =1
y+x(dy/dx) + cos(xy) (xy)' =0
y+ x(dy/dx)+ cos(xy) (y+x dy/dx) =0
( x+ xcosxy) (dy/dx) =-y -y cos(xy)
Ans: dy/dx= - {y+y cos(xy) } / {x+xcos(xy) }

3.
f(x) =3+x+e^x
f(a) =3+a+e^a =4
f ^-1 (4) =a
==> 3+a+e^a=4
        a+e^a =1
         e^a=1-a
        a= ln (1-a) => a為 y=x與 y=ln(1-x)之交點
a=0
Ans:   f ^-1 (4)=0

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