Mathematics for Quantum Mechanics: Derivatives of a Single Variable
A quick guide to derivatives with practice problems
This article is part of the series Mathematics for Quantum Mechanics.
What is a Derivative?
Geometrically speaking, given a function y(x), the derivative of the function is the slope of that function at the point x. From a physical standpoint, it is more useful to think of the slope of a curve in terms of rates of change. So the derivative of a function at a point is the instantaneous rate of change of the curve at that point. This has important physical implications and meanings in all fields of physics including quantum mechanics. As a simple example, when given a function that models the position of a particle, the derivative of that function is the velocity of the particle because the slope of the position is the velocity of the particle at that point.
Notation
There are two ways of denoting a derivative that are common in quantum mechanics. The first denotes a derivative by using a tick mark beside the function’s name. For example, the derivative of the function y(x) would be denoted as y’(x). The second notation denotes the derivative using a fraction that reference which variable we are concerned about when taking the derivative. Given a function y(x), its derivative is denoted as dy/dx, meaning that when we are taking the derivative of the function y, we are doing it with respect to the variable x. This notation will make more sense when we start taking about partial derivatives later in this series.
Basic Derivatives
There are some basic derivatives that will be used very often in quantum mechanics that are worthwhile to memorize. This will save you from having to look them up every time they are needed.
Polynomials
There is a very basic rule for taking the derivative of a polynomial. Given the function y(x) = xⁿ, its derivative is simply y’(x) = nxⁿ⁻¹. So the derivative of x² is 2x, the derivative of x³ is 2x² and so on. You can also take the derivatives of negative exponents this way as well, just remember bring down the minus sign since it is apart of the exponent. So the derivative of x⁻¹ is (-1)x⁻² = -x⁻². Remember that x⁻¹ = 1/x, x⁻² = 1/x² and so on. Note that because of this the derivative of a constant is zero. The constant 5 can be written as 5x⁰, since x⁰ is 1, but the derivative of x⁰ is 0x⁻¹ = 0.
Trigonometric Functions
Unfortunately trigonometric functions do not have a nice, compact formula for calculating the derivative, so its easier to just memorize the ones you will be using most often. The below table lists the derivatives for common trigonometric functions:
It may also be useful to know the derivatives of the hyperbolic functions (sinh, cosh, etc.) but they follow a similar pattern to the trigonometric derivatives
Exponentials and Logarithms
Exponentials and logarithms are very common in quantum mechanics , but luckily the base functions have very simple derivatives. For y(x) = eˣ, then y’(x) = eˣ, the same function! And for the natural logarithm, if y(x) = ln(x), then y’(x) = 1/x.
Product and Quotient Rules
Some functions can be written as the product of two basic functions, such as y(x) = xsin(x). These compound functions have the general form y(x) = f(x)g(x). There are also functions that can be written as the quotient of two basic functions, with the compound function having the form y(x) = f(x)/g(x). The derivatives of these types of functions can still be taken, but it is a little more complicated than taking the derivative of a basic function. To take the derivative of a function that is made up of the product of two basic function, you have to use the product rule:
In a similar fashion, to take the derivative of a function that is made up of the quotient of two basic function, you have to use the quotient rule:
A few examples using the product and quotient rules are shown below. Make sure you can work through each example yourself before moving on.
Derivatives and Sums
To take the derivative of a function that is made up of the sum of other functions, simply derive each function separately and add the results together. So written in function notion, given the function y(x) = f(x) + g(x) + h(x) + …, the derivative of y(x) would be y’(x) = f’(x) + g’(x) + h’(x) + … .
A couple examples of this concept are shown below. Make sure you can work through each example yourself before moving on.
Chain Rule
The chain rule allows you to take the derivative of a nested function. Like functions that are the products or quotients of basic functions, nested functions have to be derived using a special rule. The derivative for a function of the form y(x) = f(g(x)) is y’(x) = f’(g(x))g’(x). This is the basic form of the chain rule, so called because you have to chain the derivative of the inner function onto the derivative of the outer function.
A few examples using the chain rule are shown below. Make sure you can work through each example yourself before moving on.
Higher Order Derivatives
Up to this point we have only looked at first order derivatives, meaning that we have only taken the derivative of our initial function once. However, there are other types of derivatives, known as higher order derivatives, that are also important in the study of quantum mechanics. A second order derivative means that you take the derivative of the initial function twice (take the derivative of the initial function and then the derivative of the derivative), a third order derivative means that you derive the initial function three times, and so on. All of the above rules apply with taking higher order derivatives as well.
A few examples of calculating first, second, and third order derivatives are shown below. Make sure you can work through each example yourself before moving on.
Tips
- When you get the answer for your derivative see if it can be simplified using algebraic manipulation or trigonometric identities. For example x⁵/x³ can be simplified to x² and 2cos(x)sin(x) can be simplified using a trigonometric identity to sin(2x). Keeping your answers in their most simplified form will greatly reduce the complexity of using the answer in future problems.
- For complicated functions, try to break them down into smaller pieces that you know how to deal with. For example:
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