Second derivative

In physics and mathematics (particularly in differential calculus), the second derivative is the derivative of the derivative of a function, say [math]\displaystyle{ f(x) }[/math]. 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

[math]\displaystyle{ \frac{\mathrm{d}^2f}{\mathrm{d}x^2} }[/math], [math]\displaystyle{ \frac{\mathrm{d}^2y}{\mathrm{d}x^2} }[/math] or [math]\displaystyle{ f''(x) }[/math]

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 [math]\displaystyle{ f(x)=x^n }[/math], then the second derivative of the function can be found using the power rule for second derivatives. That is,

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

Related pages

• Second derivative test