## Quick Reference

If the function *f* is differentiable on an interval, its derived function *f*′ is defined. If *f*′ is also differentiable, then the derived function of this, denoted by *f*″, is the second derived function of *f*; its value at *x*, denoted by *f*″(*x*), or *d*^{2}*f*/*dx*^{2}, is the second derivative of *f* at *x*. (The term ‘second derivative’ may be used loosely also for the second derived function *f*″.)

Similarly, if *f*″ is differentiable, then *f*‴(*x*) or *d*^{3}*f*/*dx*^{3}, the third derivative of *f* at *x*, can be formed, and so on. The *n*-th derivative of *f* at *x* is denoted by *f*^{(n)}(*x*) or *d*^{n}*f*/*dx*^{n}. The *n*-th derivatives, for *n*≥2, are called the higher derivatives of *f*. When *y*=*f*(*x*), the higher derivatives may be denoted by *d*^{2}*y*/*dx*^{2},…, *d*^{n}*y*/*dx*^{n} or *y*″, *y*‴,…, *y*^{(n)}. If, with a different notation, *x* is a function of *t* and the derivative *dx*/*dt* is denoted by ẋ the second derivative *d*^{2}*x*/*dt*^{2} is denoted by ẍ

*Subjects:*
Mathematics.