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)
