Calculus/Higher Order Derivatives

The second derivative, or second order derivative, is the derivative of the derivative of a function. The derivative of the function $f(x)$ 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 function $f$ is defined as a function representing the slope of function $f$ , 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.

Notation

Let $f(x)$ be a function in terms of $x$ . The following are notations for higher order derivatives.

2nd Derivative	3rd Derivative	4th Derivative	$n$-th Derivative	Notes
$f''(x)$	$f'''(x)$	$f^{(4)}(x)$	$f^{(n)}(x)$	Probably the most common notation.
${\frac {d^{2}f}{dx^{2}}}$	${\frac {d^{3}f}{dx^{3}}}$	${\frac {d^{4}f}{dx^{4}}}$	${\frac {d^{n}f}{dx^{n}}}$	Leibniz notation
${\frac {d^{2}}{dx^{2}}}[f(x)]$	${\frac {d^{3}}{dx^{3}}}[f(x)]$	${\frac {d^{4}}{dx^{4}}}[f(x)]$	${\frac {d^{n}}{dx^{n}}}[f(x)]$	Another form of Leibniz notation
$D^{2}f$	$D^{3}f$	$D^{4}f$	$D^{n}f$	Euler's notation

Warning: You should not write $f^{n}(x)$ to indicate the $n$-th derivative, as this is easily confused with the quantity $f(x)$ all raised to the nth power.

The Leibniz notation, which is useful because of its precision, follows from

${\frac {d}{dx}}\left({\frac {df}{dx}}\right)={\frac {d^{2}f}{dx^{2}}}$

Newton's dot notation extends to the second derivative, ${\ddot {y}}$ , but typically no further in the applications where this notation is common.

Examples

Example 1

Find the third derivative of $f(x)=4x^{5}+6x^{3}+2x+1$ with respect to $x$.

Repeatedly apply the Power Rule to find the derivatives.

• $f'(x)=20x^{4}+18x^{2}+2$
• $f''(x)=80x^{3}+36x$
• $f'''(x)=240x^{2}+36$

Example 2

Find the 3rd derivative of $f(x)=12\sin(x)+{\frac {1}{x+2}}+2x$ with respect to $x$.

• $f'(x)=12\cos(x)-{\frac {1}{(x+2)^{2}}}+2$
• $f''(x)=-12\sin(x)+{\frac {2}{(x+2)^{3}}}$
• $f'''(x)=-12\cos(x)-{\frac {6}{(x+2)^{4}}}$

Applications

For applications of the second derivative in finding a curve's concavity and points of inflection, see "Extrema and Points of Inflection" and "Extreme Value Theorem". For applications of higher order derivatives in physics, see the "Kinematics" section.