幾題關於微分學的問題

Q1: (a, b)=(2c, - c^2)
f'(c)存在=> f'(c+)= f'(c-) => a= 2c
& f(c+)=f(c-)=f(c) => c^2= 2c^2+b => b= - c^2
Q2:
lim(h->0) (sinh)^3 / (3h) = lim(h->0) (sinh)/ h * h^2 /3 = 1*0 =0
前兩項= 1/3 f'(x) ( f(x)= (3- x sinx)^3 )
= 1/3 * 3(3- x sinx)^2 *(0 - sinx - x cosx)
= - (sinx+ xcosx)(3- x sinx)^2 ---- Ans
Q2':
By chain rule:
F'(x)= f'(y)* dy/dx ( y = (x-1)/(x+2) )
= [(x-1)/(x+2)]^3 * 3/(x+2)^2
= 3 (x-1)^3 / (x+2)^5
Note: F(x)之微分= F'(x) dx
F(x)之導函數=F'(x)
Q3:
d/dx {f[(e^x -1)/(e^x + 1)]}
= f'[(e^x -1)/(e^x +1) ] * d/dx [ (e^x -1)/(e^x +1) ]
= f'[....] *2 e^x /(e^x +1)^2
令 x=0得 f'(0)* 2*1 / 4= 2
Q4:
隱函數之導函數:
d/dx (x^3+y^3) = d/dx (3axy)
=>3x^2 + 3y^2 * y' = 3ay+ 3axy'
=> (y^2 - ax)y'= ay- x^2
=> dy/dx = y'= (ay-x^2)/(y^2 - ax)
when (x, y)=(3a/2, 3a/2) => dy/dx = - 1
tangent line is y- 3a/2 = - (x- 3a/2)
Q5:
隱函數之導函數:
d/dx (1+ xy) = d/dx [ exp(xy)]
y+ xy'= exp(xy) (y+ xy')
若 xy=0 => dy/dx 不存在 (saddle point)
若 xy≠0 => y+x y'=0 => dy/dx = - y/x

圖都掛了看不到