Higher Derivatives

The second derivative, or second order derivative, is the derivative of the derivative of a function. The derivative of the function may be denoted by $f'(x)$, and its double (or "second") derivative is denoted by $f''(x)$. This is read as "$f$ double prime of $x$," or "the second derivative of $f(x)$. " Because the derivative of a function is defined as a function representing the slope of function, the double derivative is the function representing the slope of the first derivative function.

Furthermore, the third derivative is the derivative of the derivative of the derivative of a function, which can be represented by $f'''(x)$. This is read as "$f$ triple prime of $x$", or "the third derivative of $f(x)$.". This can continue as long as the resulting derivative is itself differentiable, with the fourth derivative, the fifth derivative, and so on. Any derivative beyond the first derivative can be referred to as a higher order derivative.

If $x(t)$ represents the position of an object at time $t$, then the higher-order derivatives of $x$ have physical interpretations. The second derivative of $x$ is the derivative of $x'(t)$, the velocity, and by definition is the object's acceleration. The third derivative of $x$ is defined to be the jerk, and the fourth derivative is defined to be the jounce. A function $f$ need not have a derivative—for example, if it is not continuous. Similarly, even if $f$ does have a derivative, it may not have a second derivative. See for a graphical illustration of higher derivatives in physics.

Acceleration

Acceleration is the time-rate of change of velocity, and the second-order rate of change of position.

An example of a function with higher-order derivatives is:

$f(x) = 5x^3 + 3x^2 - x + 4$

where the higher derivatives are found to be:

• $f'(x) = 15x^2 + 6x - 1$
• $f''(x) = 30x + 6$
• $f'''(x) = 30$