微积分求导

更新时间2018-10-23 08:51:14

f(x)=(3-x^0.5)^-1,则f''(4)=多少

f'(x) = -(3 - √x)^(-2) * [-x^(-1/2)] = [x^(-1/2)](3 - √x)^(-2)；

f''(x) = [x^(-1/2)]'(3 - √x)^(-2) + [x^(-1/2)][(3 - √x)^(-2)]'

=  (-1/2)[x^(-3/2)] (3 - √x)^(-2) + [x^(-1/2)] (-2) [(3 - √x)^(-3)] (-1/2) [x^(-1/2)]

=  -1/[ 2x^(3/2) (9 - 6√x + x) ] + 1/[ x(3 - √x)^3 ]

= 1/[ x(3 - √x)^3 ] - 1/[ 18x^(3/2) - 12x^2 + 2x^(5/2) ] 

f''(4) = 1/[ 4(3 - √4)^3 ] - 1/[ 18 * 4^(3/2) - 12 * 4^2 + 2 * 4^(5/2) ] 

= 1/4 - 1/16 = 3/16 。