Second and third derivatives of a power function

Rewrite $f(x) = x^{-3}$. Differentiating once using the power rule gives $f'(x) = -3x^{-4}$. Differentiating again, multiply the exponent by the current coefficient and reduce the exponent by 1: $f''(x) = -3 \cdot (-4) x^{-5} = 12x^{-5}$.

Differentiate once more to get $f'''(x)$: multiply the coefficient $12$ by the exponent $-5$ and reduce the exponent by 1, giving $f'''(x) = 12 \cdot (-5) x^{-6} = -60x^{-6}$. So the pattern is $f''(x) = 12x^{-5}$ and $f'''(x) = -60x^{-6}$.

The third derivative is negative, since f'''(x) equals negative sixty times x to the negative sixth power. For any positive x, x to the negative sixth power is positive, so multiplying by negative sixty makes the whole expression negative. This matches the coefficient pattern seen when repeatedly applying the power rule: the signs alternate with each differentiation.