For first derivative of #f(x)=cos(-1/x^3)#, we may use function of function formula i.e. #(df(g(x)))/dx=(df)/(dg)xx(dg)/(dx)#

#(df)/(dx)=-sin(-1/x^3)xx-(-3)x^-4=-3x^-4sin(-1/x^3)# using #f(x^-n)=-nx^-(n+1)#

For second derivative, we use the formula of product derivatives i.e.

#(df(x)g(x))/(dx)=f(x)g'(x)+g(x)f'(x)#, where #f'(x)# and #g'(x)# are first derivatives of #f(x)# and #g(x)# respectively. Second derivative is mentioned as #f''(x)# and #g''(x)#.

In the given case,

#f''(x)=-3d/dx(x^-4sin(-1/x^3))=-3{x^(-4)((-3)/(x^4)cos(-1/x^3)-(-4/(x^5)sin(-1/x^3)}# or

#f''(x)=9/x^8cos(-1/x^3)-12/x^5sin(-1/x^3)#