Derivative Notation

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Objectives
  • See both ways we commonly notate derivatives, the "prime" and the "differential" notations
  • Understand why each notation has unique applications
  • Know the proper way to use each notation
Lesson Description

There are two ways to write derivatives using math symbols. A derivative is a derivative, but while each way means the same thing, some derivative applications are easier to communicate with one versus the other. This lesson will explain what each way looks like and help you understand why we might favor one over the other depending on the situation.

 
Practice Problems

Practice problems and worksheet coming soon!

 

Po-tay-to, Po-tah-to (Sort Of)

There are two ways to notate a derivative in the study of single-variable Calculus, and while they each carry the same exact meaning, one is occasionally much more convenient to use than the other, depending on what you're doing.Before looking at each one, it's worth repeating - these two symbol conventions are not conceptually different. A derivative is a derivative. Whether we use one way to write it or the other is often a matter of preference, and occasionally, serious convenience.

Prime Notation

Prime notation is by far the most common overall, since it is based on the way we express functions, and more often than not, we are working with functions in Calculus. It is almost certainly the one you were exposed to first.
Define: Prime NotationLet $f(x)$ represent a single variable differentiable function. We say that its derivative function is $f'(x)$
Because it is based on the concept of functions, we will often prefer this notation for applications where we are working directly with functions, or ones where we need to evaluate the derivative at a specified point. For example, if the expression $g'(x)$ is the derivative of some function $g(x)$, then the notation $g'(4)$ refers to the instantaneous slope of $g$ at $x=4$.

Differential Notation

The other derivative notation has its root in the classic interpretation of derivatives: slope. Differential Notation can be used for functions or standalone expressions, and uses a fraction to show the "rise over run" type behavior that single-variable derivatives signify.This notation acts like an operator, much the same way that the classic arithmetic symbols operate. It says, "we are going to take the derivative", just like $3 \times 5$ says "we are going to multiply $3$ and $5$".
Define: Differential NotationWhen placed in front of an expression, the symbol$$\frac{d}{dx}$$dictates the derivative of that expression. For example,$$\frac{d}{dx} \, x^3 = 3x^2$$When placed in front of a function or single variable it carries the same meaning, but the notation is often written short hand:$$\frac{d}{dx} f(x) \longrightarrow \frac{df}{dx}$$or$$y = x^3$$$$\Rightarrow \frac{dy}{dx} = 3x^2$$
The interpretation of this notation is that of a slope ratio. $dy/dx$ means "how much is $y$ changing per unit change in $x$?"We won't often need to try and assign such a literal interpretation to this notation but it is occasionally very helpful to understand where it comes from. Your main job is to recognize and interpret this notation when you encounter it.Common uses for differential notation include standalone expressions which are not named functions (e.g. 3x^2 - 2x +1 instead of $f(x) = 3x^2 - 2x + 1$), application problems with multi-variable formulas, and intuitive proofs of certain staple mechanics in Calculus such as the Chain Rule » or U-Substitutions ».
Vocab FYI
Differential Notation is sometimes referred to as Leibniz Notation, named after the German mathematician Gottfried Leibniz who is credited with first using it in the 1600's. Some professors will actually call it as such, even though it seems to be more and more common lately to stick with "differential notation".

Higher Derivatives

Each notation convention has its own way of communicating higher derivatives (a.k.a. multiple derivatives).For prime notation, the consensus is that multiple primes should be used up to three or four. Derivatives with a higher order should instead use a small number wrapped in parenthesis. For example, here are the first 5 derivatives of a function $f(x)$ using prime notation.Note that some folks prefer to start the parenthesis numbering at $4$, and some at $5$, but it shouldn't be a big deal - anyone would know what either symbol meant. We also don't often take derivatives of such large order.For differential notation, the convention is a little more annoying, because it's easier to screw up. The derivative differential symbol on its own is $d/dx$, which is something we should keep in mind to help us understand and remember how higher derivatives work with this notion.The derivative order number should be written as an exponent in two places: once above the $d$ in the numerator, and once above the variable in the denominator.For example, the first three derivatives of $y$ using differential notation are as follows:$$\frac{dy}{dx}$$$$\frac{d^2 y}{dx^2}$$$$\frac{d^3 y}{dx^3}$$It seems odd, but that's how it is.
Fun Fact
Because he is often called "the father of Calculus", many people assume that Isaac Newton invented the prime notation that we commonly use for derivatives. It was in fact another prominent contributor to Calculus, Joseph-Louis Lagrange (a name you'll see a bit more in multivariable Calculus), who is credited with first publishing the prime notation.In fact, Newton's own notation gained no traction, and I have never seen it in any text book. He used dots over the function or variable to show derivatives, and multiple dots to show higher derivatives.

Specific Derivatives

Sometimes in differential calculus, we don't just want to find a derivative expression, but actually want to evaluate the derivative at a specific point (which is the instantaneous tangent slope of the function at that point). This is possible to notate with either notation, but more commonly done with the prime notation since it already utilizes function notation.To do this with differential notation, which is only occasionally required, we place a vertical evaluation bar along the right side of the derivative symbol and state what the value of the variable is.$$\frac{dy}{dx} \,\, \bigg\rvert^{x=3}$$
Remember!
Don't let any teacher or professor tell you that one notation can always be preferred over the other. We must know both because there are derivative applications that lend themselves well to differential notation but are awkward or even nearly impossible to notate using prime notation, and of course, vice versa. Comfortability with both is essential to success in Calculus.

Put It To The Test

While this isn't something you'll be test on solely, this is a good opportunity to try a few simple questions that make sure you understand the difference between the two, and start to get a feel for which notation is best in which scenarios.
 
Example 1The derivative of function $g(x)$ is $8\sin(2x)$. Write this math statement using both sets of derivative notation.
Show solution
$\blacktriangleright$ Using prime notation, we would say$$g'(x) = 8\sin(2x)$$Using differential notation,$$\frac{d}{dx} \, g(x) = 8\sin(2x)$$
 
Example 2We need to take the derivative of a standalone expression, $5x - e^x$. Which notation would you use and why?
Show solution
$\blacktriangleright$ It can be confusing to use the prime notation in this case because this expression doesn't have a name. It's probably easier to write it using differential notation:$$\frac{d}{dx} \, \left[ 5x - e^x \right]$$If you were absolutely required to use prime notation for some reason, you should write it as$$\left[ 5x - e^x \right]'$$
 
Example 3We need to take the derivative of $u = \sqrt{9-x^2}$. Which notation would you use and why?
Show solution
$\blacktriangleright$ There really isn't a strong reason that one or the other is more convenient here. Both notations are fine.$$u'$$or$$\frac{du}{dx}$$
 
Example 4The volume of a cylinder is growing over time because its radius and height are each growing. The relationship is $V = \pi r^2 h$. How should we write the derivative of $V$ with respect to time?
Show solution
$\blacktriangleright$ I would choose differential here, because prime notation may leave us wondering what the real variable of this situation is. It's not $r$ or $h$, but time.$$\frac{dV}{dt}$$At this point you probably haven't yet seen time and rates word problems, but this is an entire topic you'll study once you learn more about how to find derivatives.
 
Example 5Express the derivative of $f(x)$ at the point where $x = 7$.
Show solution
$\blacktriangleright$ This is a case when it's strongly more convenient to use the prime notation.$$f'(7)$$If you were forced to use differential notation for some reason, you could express this as$$\frac{df}{dx} \,\, \Bigg|^{x=7}$$
 
Lesson Takeaways
  • Know the difference between the two ways that derivatives can be communicated
  • Be prepared to read either one and understand that both mean essentially the same thing

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