# Second derivative

In physics and mathematics (particularly in differential calculus), the second derivative is the derivative of the derivative of a function, say ${\displaystyle f(x)}$. The second derivative has many real-life applications, especially in the study of mathematical optimization and acceleration. Second derivatives can be notated in several ways, some of which are

${\displaystyle {\frac {\mathrm {d} ^{2}f}{\mathrm {d} x^{2}}}}$, ${\displaystyle {\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}}$ or ${\displaystyle f''(x)}$[1][2][3]

Other notations are used, but the above two are the most commonly used.

## Power Rule for Finding the Second Derivative

If we have a function ${\displaystyle f(x)=x^{n}}$, then the second derivative of the function can be found using the power rule for second derivatives. That is,

${\displaystyle {\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}[x^{n}]=n(n-1)x^{n-2}=(n^{2}-n)x^{n-2}}$

## References

1. "List of Calculus and Analysis Symbols". Math Vault. 2020-05-11. Retrieved 2020-09-16.
2. "Second Derivatives". Math24. Retrieved 2020-09-16.
3. "Content - The second derivative". amsi.org.au. Retrieved 2020-09-16.