D - Higher Order Derivatives Lesson

Higher Order Derivatives

If f is a differentiable function, then its derivative f' is also a function. If f' is also differentiable, then it is possible to differentiate f' to obtain a new function of x denoted by f". This new function is called the second derivative of f because it is the derivative of the derivative of f. Similarly, the third derivative f'" is the derivative of the second derivative. Continuing this process produces fourth, fifth, and subsequent higher order derivatives. Derivatives of derivatives are referred to as higher order derivatives.

Just as the first derivative f' can be interpreted as the rate of change or the slope of the curve f(x), the second derivative can be interpreted as the rate of change or the slope of the curve f', or more simply as the rate of change of a rate of change.

Notation

Several notations for higher order derivatives of a function y = f(x) are common.

Second Order Derivatives	Third Order Derivatives	nth order (4th or higher) Derivatives
y'' (y double prime)	y''' (y triple prime)	$y^n$ (nth derivative of y)
f''(x) (f double prime of x)	f'''(x) (f triple prime of x)	$LaTeX: f^{\left(n\right)}$ $f^{\left(n\right)}$ (nth derivative of f)
$LaTeX: \frac{d^2y}{dx^2}$ $\frac{d^2y}{dx^2}$ (d two y d x two or the second derivative of y with respect to x both times)	$LaTeX: \frac{d^3y}{dx^3}$ $\frac{d^3y}{dx^3}$ (d three y d x three or the third derivative of y with respect to x three times)	$LaTeX: \frac{d^ny}{dx^n}$ $\frac{d^ny}{dx^n}$ (dnydxn or the nth derivative of y with respect to x n times)
$LaTeX: D_x^2\left(f\right)$ $D_x^2\left(f\right)$ (D2 sub x of f or the second derivative of f with respect to x both times)	$LaTeX: D_x^3\left(f\right)$ $D_x^3\left(f\right)$ (D3 sub x of f or the third derivative of f with respect to x three times)	$LaTeX: D_x^n\left(f\right)$ $D_x^n\left(f\right)$ (Dn sub x of f or the nth derivative of f with respect to x n times)

Second Order Derivatives

Third Order Derivatives

nth order (4th or higher) Derivatives

y''

(y double prime)

y'''

(y triple prime)

equation image indicator LaTeX: y^n $y^n$