The Derivative as a Function: Learn It 4

Higher-Order Derivatives

The derivative of a function is itself a function, so we can find the derivative of a derivative. For example, the derivative of a position function is the rate of change of position, or velocity. The derivative of velocity is the rate of change of velocity, which is acceleration. The new function obtained by differentiating the derivative is called the second derivative.

Furthermore, we can continue to take derivatives to obtain the third derivative, fourth derivative, and so on. Collectively, these are referred to as higher-order derivatives. The notation for the higher-order derivatives of y=f(x)y=f(x) can be expressed in any of the following forms:

f(x),f(x),f(4)(x),,f(n)(x)f′′(x),f′′′(x),f(4)(x),,f(n)(x)
y,y,y(4),,y(n)y′′,y′′′,y(4),,y(n)
d2ydx2,d3ydx3,d4ydx4,,dnydxnd2ydx2,d3ydx3,d4ydx4,,dnydxn

It is interesting to note that the notation for d2ydx2d2ydx2 may be viewed as an attempt to express ddx(dydx)ddx(dydx) more compactly. Analogously,

ddx(ddx(dydx))=ddx(d2ydx2)=d3ydx3ddx(ddx(dydx))=ddx(d2ydx2)=d3ydx3

higher-order derivatives

Higher-order derivatives are the derivatives of a function taken multiple times. The first derivative is the rate of change of the function, the second derivative is the rate of change of the first derivative, and so on.

 

Notation: Common notations for higher-order derivatives include:

f(x),f(x),f(4)(x),,f(n)(x)f′′(x),f′′′(x),f(4)(x),,f(n)(x)
y,y,y(4),,y(n)y′′,y′′′,y(4),,y(n)
d2ydx2,d3ydx3,d4ydx4,,dnydxnd2ydx2,d3ydx3,d4ydx4,,dnydxn

For f(x)=2x23x+1f(x)=2x23x+1, find f(x)f′′(x).

The position of a particle along a coordinate axis at time tt (in seconds) is given by s(t)=3t24t+1s(t)=3t24t+1 (in meters). Find the function that describes its acceleration at time tt.