Differentiate the following functions with respect to x.
1a. ln(ln(ln x ))
b. x^sinx
c. ln (e^x+2x+1/e^x-3x-1)
2.Find dy/dx if
x=a cos 2 u, y=cos u+1
3a. limx->∞ ln(1+e^x)/1+x
b.lim x->0 2x - sin2x / x - sinx
c.limx->0 (1/x - 1/sinx)
d.lim x->0+ x ln x
e.lim x->∞ (cos 1/x)^x
4.If f(x)= x/x^2-1, find f^(n) (0)
for every n€N.
5. A reservoir has the shape of a right circular cone. The altitude is 10 metres, and the radius of the base is 4 metres. Water is poured into the reservoir at a constant rate of 5 cabic metres per minute. How fast is the water level rising when the depth of the water is 5 metres, and if
(a) the vertex of the cone is up?
(b) the vertex of the cone is down?
更新1:
3a. limx=>無限 ; ln(1+e^x)/1+x b.lim x=>0 ;0 2x - sin2x / x - sinx c.limx=>0 (1/x - 1/sinx) d.lim x=>0 + ; x ln x e.lim x>無限; (cos 1/x)^x 4.If f(x)= x/x^2-1, find f^(n) (0) for every n 包含N.
