Higher derivative

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Definition

Terminology

Higher derivatives are also called repeated derivatives or iterated derivatives.

Function and prime notation

Suppose f is a function and k is a nonnegative integer. The k^{th} derivative of f, denoted f^{(k)} or f^{'''\dots'} where {}^' occurs a total of k times, is defined as the function obtained by differentiating f a total of k times (i.e., taking the derivative, then taking the derivative of that, and so on, a total of k times). The first few cases are shown explicitly:

Value of k Notation with repeated primes for f^{(k)} f^{(k)} notation Definition In words
0 \! f \! f^{(0)} \! f the original function
1 \! f' \! f^{(1)} \! f' the derivative, also called the first derivative
2 \! f'' \! f^{(2)} \! (f')^' the second derivative
3 \! f''' \! f^{(3)} \! ((f')^')^' the third derivative

We could also define the k^{th} derivative inductively as:

\! f^{(k)} = (f^{(k-1)})^'

or as:

\! f^{(k)} = (f')^{(k-1)}

with the base case f^{(0)} = f.

Leibniz notation

Suppose y = f(x), so y is a dependent variable depending on x, the independent variable. The k^{th} derivative of y with respect to x is denoted:

\! \frac{d^k}{(dx)^k}y

or as:

\! \frac{d^ky}{dx^k}

and is defined as:

\! \frac{d^ky}{dx^k} = \frac{d}{dx}\left[\frac{d}{dx} \left[ \dots \left[\frac{d}{dx}(y)\right] \dots \right] \right]

where the d/dx occurs k times. Alternatively we can define it inductively as:

\! \frac{d^ky}{dx^k} = \frac{d}{dx} \frac{d^{k-1}y}{dx^{k-1}}

with the base case k = 0 being defined as \frac{d^0y}{dx^0} = y.