Concept preview

What you will be able to do

1. 01Define the higher-order derivatives $f''$, $f'''$, and $f^{(n)}$ as repeated differentiation — each one the slope (derivative) function of the previous — and read the notations $\dfrac{d^2y}{dx^2}$ and $\dfrac{d^ny}{dx^n}$.
2. 02Compute the second and third derivatives of polynomials and simple composites accurately, keeping arithmetic under control across successive passes.
3. 03Recognise patterns in repeated derivatives (for example the period-4 cycle of $\sin x$) and interpret the second derivative as concavity and as acceleration.

Start with a function $f$. Its derivative $f'$ is a brand-new function that reports the slope of $f$ at every input. Because $f'$ is a function, we may differentiate it too; the result is the SECOND derivative, written $f''$ (read "f double prime"). Differentiating once more gives the THIRD derivative $f'''$, and in general the $n$-th derivative $f^{(n)}$, where beyond the third we switch to the parenthesised-number notation because writing four or five prime marks is unreadable. The rule is mechanical: to get the next derivative, differentiate the current one. There is no separate "second-derivative rule" — you just apply your ordinary rules again to the function you last produced.