Introduction
In calculus, taking the derivative of a function allows us to understand the rate of change of a function. This process allows us to gain information about the nature of a function, giving us a variety of applications to different problems. When we want to take the derivative of a function multiple times, we refer to higher order differentiation.
Higher Order Differentiation
Let’s review the notation for the first derivative of any function $f(x)$. Unfortunately, this doesn’t have a uniform representation, and you might encounter the following forms:
$$f'(x), \quad \dfrac{df(x)}{dx}, \quad \dfrac{d}{dx}f(x). $$
Higher-order differentiation is a process where we take the derivative of a function multiple times. Let’s consider the second derivative, which has the following different representations:
$$f”(x), \quad \dfrac{d^2f(x)}{dx^2}, \quad \dfrac{d^2}{dx^2}f(x).$$
We can go further and take third, fourth, fifth derivatives, and so on. In general, the formula for the $n$’th derivative is:
$$f^{(n)}(x), \quad \dfrac{d^nf(x)}{dx^n}, \quad \dfrac{d^n}{dx^n}{f(x)}.$$
Examples
Suppose $f(x) = x^5 + 3x^2 + 1$. Let’s find $\dfrac{d^2}{dx^2}f(x)$, or the second derivative. Notice that the second derivative is just the derivative of the first derivative:
$$\dfrac{d^2}{dx^2}(f(x)) = \dfrac{d}{dx}\left(\dfrac{d}{dx}f(x)\right).$$
Using the Power Rule,
\[\begin{align*} \dfrac{d^2}{dx^2}(f(x)) &= \dfrac{d}{dx}\left(\dfrac{d}{dx}f(x)\right) \\ &= \dfrac{d}{dx}\left(\dfrac{d}{dx}(x^5 + 3x^2 + 1)\right) \\ &= \dfrac{d}{dx}(5x^4 + 6x) \\ &= \boxed{20x^3 + 6}. \end{align*}\]
Let’s try another example. Suppose $g(x) = x^6 – 2x^3 + 2x$. Let’s find the third derivative, or $\dfrac{d^3}{dx^3}g(x)$. Using the Power Rule,
\[\begin{align*} \dfrac{d^3}{dx^3}(g(x)) &= \dfrac{d^2}{dx^2}\left(\dfrac{d}{dx}g(x)\right) \\ &= \dfrac{d^2}{dx^2} \left( \dfrac{d}{dx} (x^6 – 2x^3 + 2x) \right) \\ &= \dfrac{d^2}{dx^2}(6x^5 – 6x^2 + 2) \\ &= \dfrac{d}{dx}\left(\dfrac{d}{dx} (6x^5 – 6x^2 + 2)\right) \\ &= \dfrac{d}{dx}(30x^4 – 12x) \\ &= \boxed{120x^3 – 12}. \end{align*}\]
The concept of higher order derivatives has useful applications in other fields. In physics, the second order derivative of position is known as acceleration. For more information about derivatives, check out other blogs on iacedcalculus.com.
