Higher Order Derivative

In calculus, the concept of derivatives is fundamental. The derivative of a function represents the rate at which the function's value changes with respect to a change in its input value. When we take the derivative of a function, we get another function that can also be differentiated if the original function is sufficiently smooth. Repeatedly differentiating a function gives rise to higher order derivatives.

First Order Derivative

The first order derivative of a function $f(x)$ is denoted as $f'(x)$ or $\frac{df}{dx}$ and represents the instantaneous rate of change of the function with respect to $x$. It is defined as the limit:

$$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$

Higher Order Derivatives

The second order derivative is the derivative of the first order derivative and is denoted as $f''(x)$ or $\frac{d^2f}{dx^2}$. Similarly, the third order derivative is the derivative of the second order derivative, and so on. The $n$-th order derivative is denoted as $f^{(n)}(x)$ or $\frac{d^nf}{dx^n}$.

The higher order derivatives can be interpreted in various ways depending on the context, such as the rate of change of the rate of change for the second order derivative, or the "acceleration" of the function's graph.

Table of Differences and Important Points

Order	Notation	Interpretation	Example
1st	$f'(x)$ or $\frac{df}{dx}$	Instantaneous rate of change of $f$ with respect to $x$	If $f(x) = x^2$, then $f'(x) = 2x$
2nd	$f''(x)$ or $\frac{d^2f}{dx^2}$	Rate of change of the first derivative or "acceleration" of $f$	If $f(x) = x^2$, then $f''(x) = 2$
3rd	$f^{(3)}(x)$ or $\frac{d^3f}{dx^3}$	Rate of change of the second derivative	If $f(x) = x^2$, then $f^{(3)}(x) = 0$
nth	$f^{(n)}(x)$ or $\frac{d^nf}{dx^n}$	Rate of change of the $(n-1)$-th derivative	If $f(x) = x^n$, then $f^{(n)}(x) = n!$

Formulas

For a function $f(x)$, the higher order derivatives are obtained by repeated application of the differentiation rules. Some common rules include:

• Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
• Product Rule: If $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$.
• Quotient Rule: If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$.
• Chain Rule: If $f(x) = g(h(x))$, then $f'(x) = g'(h(x))h'(x)$.

Examples

Let's go through a few examples to illustrate the concept of higher order derivatives.

Example 1: Polynomial Function

Consider the function $f(x) = x^3 - 3x^2 + 5x - 7$.

• First derivative: $f'(x) = 3x^2 - 6x + 5$
• Second derivative: $f''(x) = 6x - 6$
• Third derivative: $f^{(3)}(x) = 6$
• Fourth derivative and beyond: $f^{(n)}(x) = 0$ for $n \geq 4$

Example 2: Trigonometric Function

Consider the function $f(x) = \sin(x)$.

• First derivative: $f'(x) = \cos(x)$
• Second derivative: $f''(x) = -\sin(x)$
• Third derivative: $f^{(3)}(x) = -\cos(x)$
• Fourth derivative: $f^{(4)}(x) = \sin(x)$

Notice that for trigonometric functions like sine and cosine, the derivatives are periodic and repeat every four derivatives.

Example 3: Exponential Function

Consider the function $f(x) = e^x$.

• First derivative: $f'(x) = e^x$
• Second derivative: $f''(x) = e^x$
• Third derivative: $f^{(3)}(x) = e^x$

For the exponential function $e^x$, all higher order derivatives are the same as the original function.

Conclusion

Higher order derivatives provide deeper insights into the behavior of functions beyond just the slope of the tangent line. They are used in various fields such as physics for understanding motion (velocity and acceleration), in economics for analyzing concavity and convexity of utility functions, and in engineering for control systems and signal processing. Understanding how to compute and interpret higher order derivatives is essential for advanced studies in calculus and its applications.