Higher Order Derivatives

The derivative $\dfrac{dy}{dx}=f'(x)$ of a differentiable function is itself a function of $x$, so — if that new function is itself differentiable — it can be differentiated again, giving what is called the second derivative, written $y''$, $f''(x)$, or $\dfrac{d^2y}{dx^2}$ (in Leibniz notation, $\dfrac{d}{dx}\!\left(\dfrac{dy}{dx}\right)$). Continuing, differentiating $f''(x)$ gives the third derivative $f'''(x)=\dfrac{d^3y}{dx^3}$, and in general the $n$-th derivative $f^{(n)}(x)=\dfrac{d^ny}{dx^n}$ is obtained by differentiating $f(x)$ exactly $n$ times in a row. Collectively these are called higher order derivatives (also written $y_2,y_3,\ldots,y_n$ in an alternative shorthand). By the first-principles limit definition, $f'(x)=\displaystyle\lim_{h\to0}\dfrac{f(x+h)-f(x)}h=\dfrac{dy}{dx}$ and, applying the same limit to $f'$ itself, $f''(x)=\displaystyle\lim_{h\to0}\dfrac{f'(x+h)-f'(x)}h=\dfrac{d^2y}{dx^2}$.

Worked Example. For $f(x)=x^3-x$: $f'(x)=3x^2-1$; differentiating again, $f''(x)=6x$; differentiating once more, $f'''(x)=6$ — a constant, which is the slope of the (straight-line) graph of $f''(x)=6x$. Every derivative beyond this one is $0$. …