What is the second derivative of #f(x)=cos(1/x) #?

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Answer 1 · 2018-05-28T19:01:28

#f''(x)=(-cos(1/x)-2xsin(1/x))/x^4#

Here,

#f(x)=cos(1/x)#

Diff.w.r.t. #x# , using Chain Rule :

#=>f'(x)=-sin(1/x)d/(dx)(1/x)#

#=>f'(x)=-sin(1/x)(-1/x^2)#

#=>f'(x)=sin(1/x)/x^2#

Again diff.w.r.t. #x#, using Quotient Rule :

#=>f''(x)=(x^2d/(dx)(sin(1/x))-sin(1/x)d/(dx)(x^2))/(x^2)^2#

#=>f''(x)=(x^2*cos(1/x)(-1/x^2)-sin(1/x)*2x)/x^4#

#=>f''(x)=(-cos(1/x)-2xsin(1/x))/x^4#