1.
f'(x) = lim(q->0)(f(x+q)-f(x))/q = -sinx
2.
(1) y' = -1/x^2,切线方程 y-1 = -(x-1)
(2) y' = 3x^2,切线方程 y-8 = 12(x-2)
3.
(1) y' = 6(x^3 - x)^5 * (3x^2 - 1)
(2) y' = -x^2 cos(1/x) (1/x^2) + 2xsin(1/x)
= 2xsin(1/x) - cos(1/x)
(3) y' = 1/(x+√(x^2+a^2)) * (1+1/(2√(x^2+a^2)))
= (2√(x^2+a^2)+1) / 2(x√(x^2+a^2)+(x^2+a^2))
(4) y' = 1/(1+(1-x)^2) * (-1)
= -1/(x^2-2x+2).
4.
y(n*') = (x+n)e^x
5.
(1) dy = csc(x/2)cos(x/2)/2 dx = cot(x/2)dx/2
(2) dy = (-e^(-x) cos(3-x) + e^(-x) sin(3-x))dx
(3) dy = ( 1/(1+x^2) * (1+x)/(1-x) + 2arctanx/(1-x)^2 )dx
= (1+x)(1-x)+2arctanx(1+x^2)/(1-x)^2(1+x^2) dx
(4)e^(x/y)(-xdy + ydx)/y^2)-ydx-xdy=0
6.
(1)、x立方+y立方-3xy=0;(2)、arctany/x=ln√x平方+y平方
(1) 3x^2dx + 3y^2dy - 3xdy - 3ydx = 0
(3x^2 - 3y)dx = - (3y^2 + 3x)dy
dy/dx = -(3x^2 - 3y)/(3y^2 + 3x).
(2) 1/(1+(y/x)^2)(xdy-ydx)/y^2 = 1/(√x^2+y^2) * 1/2 * (x^2+y^2)^(-1/2) * (2xdx + 2ydy)