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)

equation image indicator LaTeX: y^n

(nth derivative of y)

f''(x)

(f double prime of x)

f'''(x)

(f triple prime of x)

equation image indicator $LaTeX: f^{\left(n\right)}$

(nth derivative of f)

equation image indicator $LaTeX: \frac{d^2y}{dx^2}$

(d two y d x two or the second

derivative of y with respect to x both times)

equation image indicator $LaTeX: \frac{d^3y}{dx^3}$

(d three y d x three or the third

derivative of y with respect to x three times)

$LaTeX: \frac{d^ny}{dx^n}$

(dnydxn or the nth derivative of y with respect to x n times)

equation image indicator $LaTeX: D_x^2\left(f\right)$

(D2 sub x of f or the second

derivative of f with respect to x both times)

equation image indicator $LaTeX: D_x^3\left(f\right)$

(D3 sub x of f or the third

derivative of f with respect to x three times)

equation image indicator $LaTeX: D_x^n\left(f\right)$

(Dn sub x of f or the nth derivative of f with respect to x n times)

image of a book with list of higher order derivatives

Click HERE to participate in several self-check opportunities with a variety of self-entered problems on the higher derivative. Simply enter the derivative order followed by the expression, e.g., third derivative of sin 2x.

Higher Order Derivatives Practice

Higher Order Derivatives: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of higher order derivatives.

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