Question

9. Let g(x) = In(x)/x - 1 Then the
hundredth derivative at x = 1 is? *​

Answer 1

Step-by-step explanation:  The key here is to again expand the numerator as a Taylor series centered at x = 1

Hence we have the Taylor series as

ln(x)=(x−1)1−(x−1)22+(x−1)33…..∞

Hence our function g(x) transforms into

g(x)=1−(x−1)2−(x−1)23+(x−1)34…..∞

This now simplifies into polynomial differentiation and the hundredth derivative can be written as

g(100)(x)=100!101−101!×(x−1)102…..∞

Substituting x = 1 yields

100!⁄101.