Question regarding the derivative terminalogy and wording

In summary, when discussing calculus and taking derivatives of a function ##f(x)##, it is important to note that ##f(x)## is not a function itself, but rather the value of the function ##f## evaluated at ##x##. To avoid confusion and misuse of notation, it may be better to use the 'prime' notation, ##f'##, which clearly indicates that it is a derivative function. However, the ##\frac{d}{dx}## notation does have a clear link to the definition of the derivative as the limit of a ratio. To avoid ambiguity, it is also possible to use implied parentheses or write ##\frac{df}{dx}(x)## when taking derivatives. This issue can
  • #1
Mr Davis 97
1,462
44
When doing calculus, we typically say that we "take the derivative of a function ##f(x)##." However, rigorously, ##f(x)## is not a function but rather the value of the function ##f## evaluated at ##x##. Thus, in order for this wording to be correct shouldn't we have to write something like ##\displaystyle \frac{d}{dx}(x \mapsto f(x) )## instead of ##\displaystyle \frac{d}{dx}f(x)##, as we normally do? Something about this mix up of words and notation bothers me.
 
Mathematics news on Phys.org
  • #2
You're right. Out of the several notations available for derivatives, the ##\frac{d}{dx}## notation is the least clear and the most subject to misuse and subsequent confusion. I think the 'prime' notation is better, as ##f'## is a function that is the derivative of ##f##, so that ##f'(x)## is ##f'## evaluated at ##x##.

One advantage of the ##\frac{d}{dx}## notation though is that it has a clearer link to the definition of the derivative as the limit of a ratio.

##\frac{d}{dx}f(x)## can be OK if you infer the existence of implied parentheses in the right place, ie ##\left(\frac{d}{dx}f\right)(x)##. Another way to write it, following the same principle, is ##\frac{df}{dx}(x)##.

The muddle can get much worse when one does partial derivatives. It's possible to write horribly ambiguous and misleading formulas using fairly commonly-used notation.
 

1. What is the definition of a derivative?

A derivative is a mathematical concept that measures the rate of change of a function at a specific point. It represents the slope of the tangent line to the function at that point.

2. What is the difference between the words "derivative" and "differentiation"?

The term "derivative" refers to the mathematical concept itself, while "differentiation" is the process of finding the derivative of a function.

3. What is the notation used to represent a derivative?

The most common notation for a derivative is f'(x), where f is the function and x is the point at which the derivative is being evaluated. Other notations include dy/dx and Df(x).

4. How is the derivative of a function calculated?

The derivative of a function can be calculated using the limit definition of a derivative, which involves finding the slope of a secant line as the two points on the function get closer and closer together. Alternatively, there are rules and formulas for finding the derivative of different types of functions, such as power functions, trigonometric functions, and exponential functions.

5. What is the significance of the derivative in mathematics and science?

The derivative has many applications in mathematics and science, including optimization, physics, and economics. It allows us to analyze the rate of change of a function and make predictions about its behavior, which is crucial in understanding many real-world phenomena.

Similar threads

  • General Math
Replies
3
Views
786
Replies
2
Views
737
Replies
4
Views
362
Replies
4
Views
323
  • General Math
Replies
6
Views
1K
  • Calculus
Replies
6
Views
1K
  • Advanced Physics Homework Help
Replies
8
Views
787
  • General Math
Replies
13
Views
2K
Replies
2
Views
894
Replies
3
Views
1K
Back
Top