Second derivative

In physics and mathematics (particularly in differential calculus), the second derivative is the derivative of the derivative of a function, say $f(x)$. 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

$\frac{{\mathrm{d}}^{2}f}{\mathrm{d}{x}^2}$, $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$ or $f^{″}(x)$

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 $f(x)=x^n$, then the second derivative of the function can be found using the power rule for second derivatives. That is,

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

Related pages

• Second derivative test