Higher-order derivatives

To build the Taylor polynomial we need the higher-order derivatives: the second derivative, the third, and in general the $n$-th derivative.

Definition — $n$-th derivative

The second derivative of $f$ is the derivative of the derivative: $f''(x) = (f'(x))'$. More generally, the $n$-th derivative is obtained by differentiating $n$ times: $f^{(n)}(x)$. By convention $f^{(0)}(x) = f(x)$.

Example — Higher-order derivatives of $e^x$ and $\sin x$

Let $f(x) = e^x$. $f'(x) = e^x$, $f''(x) = e^x$, …, $f^{(n)}(x) = e^x$ for every $n$.

Let $f(x) = \sin x$. $f'(x) = \cos x$, $f''(x) = -\sin x$, $f'''(x) = -\cos x$, $f^{(4)}(x) = \sin x$, and then it repeats (cycle of period $4$).

Links

Topics: Derivatives
Concepts: Derivative · Higher-order derivative
Skills: Differentiating