What is meant by "take the derivative of a function"?

In summary, Bourbaki defines a function as an operation that associates an element x with an element y in a given relation, where y is the value of the function at x and the function is determined by the functional relation. Two equivalent functional relations determine the same function. In the conversation, the use of shorthand notation for derivatives is discussed, where f(x) is not the function itself but rather the image of x under f, and dy/dx is the rate of change of y with respect to x. The shorthand notation can sometimes cause ambiguity, but in most cases it is clear. However, not all differentiable functions are analytic, so it may be more accurate to say that a differentiation operator takes in analytic expressions and outputs analytic expressions.
  • #1
Mr Davis 97
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Bourbaki defines a function as follows: We give the name of function to the operation which associates with every element x the element y which is in the given relation with x; y is said to be the value of the function at the element x, and the function is said to be determined by the given functional relation. Two equivalent functional relations determine the same function.

This means that f(x), an analytic expression for example, is the image of x under f, and not the function itself, just as y, the output value, is not the function itself. Thus, why do we say "take the derivative of f(x)" when f(x) is not even the function, but rather the image of the variable x under f? In addition, the notation dy/dx gives the feeling that the input to a differentiation operator is not a function at all, but an expression that defines a function, y=f(x). Given that we say "take the derivative of such and such function", shouldn't we always write D(x↦x2)=(x↦2x) rather than D(x2)=2x. Is the latter just a shorthand for the more rigorous former?
 
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  • #2
Yes, it's just a shorthand. In the vast majority of cases the meaning is perfectly clear and causes no problems. But in some circumstances it can cause ambiguity, and then the author needs to take more care.
 
  • #3
andrewkirk said:
Yes, it's just a shorthand. In the vast majority of cases the meaning is perfectly clear and causes no problems. But in some circumstances it can cause ambiguity, and then the author needs to take more care.
Why don't we say that a differentiation operator takes an analytic expression (such as a combination of elementary functions) as input and returns and analytic expression as output, since that's really what the differentiation operator does? We see ##\displaystyle \frac{d}{dx}x^2 = 2x## in real calculations rather than ##\displaystyle \frac{d}{dx}(x \mapsto x^2) = (x \mapsto 2x)##
 
  • #4
Not all differentiable functions are analytic.

How would you write diff eqns under such an approach?
 
  • #5
andrewkirk said:
Not all differentiable functions are analytic.

How would you write diff eqns under such an approach?

I guess you're right. But here's what I understand: something like y = 5x or y = ln(x) describes a condition for a function, just as a table or a graph could describe a condition. y is not the function nor is 5x or ln(x). This is why I don't get why we say a derivative takes in functions and outputs functions. y is not a function, yet that is what we input into d/dx. Thus it would seem as though derivatives should be defined such that they take in expressions that define a function and output expressions that define another function. You can't calculate a derivative without an expression...
 
  • #6
Mr Davis 97 said:
y is not a function
Correct. We assign the value of the function to the variable y.
Mr Davis 97 said:
Thus it would seem as though derivatives should be defined such that they take in expressions that define a function and output expressions that define another function.
Yes. Given a function f(x), the derivative (with respect to x) is [itex] \frac{df(x)}{dx}[/itex].
 
  • #7
Svein said:
Correct. We assign the value of the function to the variable y.
Yes. Given a function f(x), the derivative (with respect to x) is [itex] \frac{df(x)}{dx}[/itex].
If y is not the function but rather the value of the function, then what does ##\frac{dy}{dx}## really mean? Why would this be the correct type of argument in the d/dx operator?
 
  • #8
Mr Davis 97 said:
If y is not the function but rather the value of the function, then what does ##\frac{dy}{dx}## really mean? Why would this be the correct type of argument in the d/dx operator?
You can think of dy/dx as the rate of change of y with respect to change in x. As andrewkirk said in an earlier post, it's just shorthand.
 

FAQ: What is meant by "take the derivative of a function"?

What is meant by "take the derivative of a function"?

The derivative of a function is a mathematical concept that represents the rate of change of that function at a specific point. It is essentially the slope of the function at that point, and it tells us how quickly the function is changing at that point.

Why is taking the derivative of a function important?

Taking the derivative of a function allows us to analyze and understand the behavior of the function. It is used in many fields of science and engineering, including physics, economics, and engineering, to model and predict the behavior of systems and processes.

How do you take the derivative of a function?

The process of taking the derivative of a function involves applying the rules of differentiation, which include the power rule, product rule, quotient rule, and chain rule. These rules allow us to find the derivative of more complex functions by breaking them down into simpler parts.

Can any function be differentiated?

Not all functions can be differentiated. Some functions, such as step functions and absolute value functions, do not have derivatives at certain points. These points are known as discontinuities, and the function is said to be non-differentiable at those points.

What is the relationship between a function and its derivative?

The derivative of a function is closely related to the original function. The derivative of a function f(x) is denoted as f'(x) or dy/dx, and it gives us information about the behavior of f(x). For example, if f'(x) is positive, the function is increasing at that point, and if it is negative, the function is decreasing at that point.

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