Why do we use"dx" in the derivative dy/dx?

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In summary: I really appreciate your response, thanks.In summary, the concept of "dx" and "dy" in calculus is not meant to be seen as numbers, but rather as a shorthand notation for a particular limit. This notation can be useful in remembering certain rules of calculus, but it is important to understand the concept of limits in order to properly interpret these symbols. The history of derivatives without limits is fascinating and serves as a reminder that calculus is constantly evolving.
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sviego
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Homework Statement
What purpose does "dx" have, and how can one solve for it?

I understand how to solve derivatives, but am confused as to why we divide by "dx" when we're clearly only solving for the derivative of "dy." Thus, couldn't one write d/dx(x^2) as simply d(x^2)?
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dx = ?
My theory is that dx = 1(x^0) = 1, which would mean d/dx(x^2) = 2(x^1)/1(x^0) = 2x/1 = 2x.

I know that the derivative is literally the change in "y" over change in "x," but am confused as to what value the change in "x" has.
 
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It does not have a value. It is a notation for a particular limit, i.e., the entire ##dy/dx## should be seen as short-hand for ##\lim_{\epsilon \to 0}[(y(x+\epsilon)-y(x))/\epsilon]##.
 
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Orodruin said:
the entire ##dy/dx## should be seen as short-hand for ##\lim_{\epsilon \to 0}[(y(x+\epsilon)-y(x))/\epsilon]##.
Or, changing your notation a bit: define ##\Delta x = (x+ \epsilon) - x = \epsilon## and ##\Delta y = y(x+\epsilon)-y(x)##. Then ##\frac {dy} {dx}## becomes a short-hand for $$\lim_{\Delta x \to 0} \frac {\Delta y} {\Delta x}$$ The notations ##dx## and ##dy## are supposed to make you think about infinitesimally tiny ##\Delta x## and ##\Delta y##.
 
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sviego said:
Homework Statement: What purpose does "dx" have, and how can one solve for it?
To repeat what others said, the modern interpretation of ##dy/dx## in a calculus course does not regard ##dy## and ##dx## as being numbers and it does not regard ##dy/dx## as being a fraction involving two numbers. If f(x) is a function, perhaps you have seen the notation f'(x) to denote the derivative of f. The modern interpretation of ##df/dx## is that it means ##f'(x)##.

There are advanced mathematical subjects where notation such ##df## is used without any denominator and there is an advanced (but not particularly popular) method of inventing a number system which contains "infinitesimals" and these are often denoted by symbols like ##dx##.

Calculus has a long history and the early inventors of calculus did not have the modern concept of limits. So they did think of ##dy/dx## as fraction.

My theory is that dx = 1(x^0) = 1, which would mean d/dx(x^2) = 2(x^1)/1(x^0) = 2x/1 = 2x.
I don't know what you mean by "d/dx(^2)". For example, is "d/dx(^2) sin(x)" supposed to have a meaning?

Before the invention of limits, the approach to derivatives went something like this:
Let ## y = x^2##. The derivative of y (in the old days) was ##\frac{dy}{dx} = \frac{(x+dx)^2 - x^2}{dx} = \frac{x^2 + 2xdx + (dx)^2 - x^2}{dx} = \frac{ 2xdx + (dx)^2}{dx} = 2x + dx ##. To get the final result ##\frac{dy}{dx} = 2x ## we are forced to think of ##dx## as a number that is effectively zero, but not causing the embarrasement of dividing by zero before we reduce the fraction. There is also the difficulty that the numerator ##(x + dx)^2 - x^2## would be zero if ##dx## was zero.

There was a struggle to interpret the above algebraic manipulations in various ways. A satisfactory method of interpreting symbols like ##dx## as a new type of number was finally developed in the 1960's . ( https://en.wikipedia.org/wiki/Non-standard_analysis ). However, long before that, the concept of limits was introduced to define the concepts of calculus in a precise manner. Typical modern calculus texts define concepts in terms of limits, not in terms of nonstandard analysis.

Notation such as ##\frac{dy}{dx}## is useful in remembering certain rules of calculus and many physics texts present arguments treating ##dy## and ##dx## as numbers. Thinking about them this way is useful in understanding physics.

For example, the "chain rule" is ##D_x f(g(x)) = f'(g(x)) g'(x)##. This can be remembered by ##\frac{df}{dx} = \frac{dy}{dg} \frac{dg}{dx}##, which treats the derivatives as fractions. However, it might lead to making the mistake ##D_x sin(x^2) = cos(x) 2x##. The other notation makes it clear that ##f'()## is to be evaluated at ##x^2##, so we get ##D_x sin(x^2) = cos(x^2) 2x##.
 
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Wow, this was really informative! Thank you Stephen.

Stephen Tashi said:
I don't know what you mean by "d/dx(^2)".

I meant "the derivative of x(^2)," but as a random example. I see now, however, that we're not trying to find the derivative of x because (and thanks to limits) the denominator for dx is simply zero.

The history of derivatives without limits is fascinating! I forget that calculus wasn't born in a single day; it's constantly shaping and evolving.
 
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Hi Orodruin! This is my first time writing on these forums, so thanks for the help.

Orodruin said:
It does not have a value. It is a notation for a particular limit, i.e., the entire ##dy/dx## should be seen as short-hand for ##\lim_{\epsilon \to 0}[(y(x+\epsilon)-y(x))/\epsilon]##.

I see, so "dx" is simply shorthand for the denominator of the limit where dx→0 when dx = h and h→0.
 
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Hi jtbell! What an interesting addition to Orodiun's response, I really appreciate it.

jtbell said:
The notations ##dx## and ##dy## are supposed to make you think about infinitesimally tiny ##\Delta x## and ##\Delta y##.

That makes so much sense! Simply put, the derivative describes the microscopic change in y and x as x approaches zero.
 
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sviego said:
Hi Orodruin! This is my first time writing on these forums, so thanks for the help.
I see, so "dx" is simply shorthand for the denominator of the limit where dx→0 when dx = h and h→0.

##h## is a number. In ##\frac{dy}{dx}## then ##dx## has no independent meaning. The whole thing means "the derivative of ##y## with respect to ##x##'. But ##dy## and ##dx## have no independent meaning.

Things in mathemetics are what you define them to be.

One of the next steps in calculus is to give meaning to ##dx## as a "differential" or "infinitesimal". But, technically, that is different from the use of ##dx## in the derivative.
 
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sviego said:
Hi jtbell! What an interesting addition to Orodiun's response, I really appreciate it.
That makes so much sense! Simply put, the derivative describes the microscopic change in y and x as x approaches zero.

As this is "calculus and beyond", "microscopic" has no mathematical meaning. The derivative is a limit, which has a well-defined mathematical meaning. The definition of a limit uses only the properties of real numbers.
 
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sviego said:
My theory is that dx = 1(x^0) = 1, which would mean d/dx(x^2) = 2(x^1)/1(x^0) = 2x/1 = 2x.
Stephen Tashi said:
I don't know what you mean by "d/dx(^2)". For example, is "d/dx(^2) sin(x)" supposed to have a meaning?
Stephen, the OP didn't write "d/dx(^2)." You quoted what the OP wrote, but apparently misread it.
 
  • #11
Ultimately, the expression f'(x)dx is called the differential* and measures the (linear) approximation to the change of the values of the function. If, say, f(x)=##x^2## then the differential. 2xdx measures changes in a neighborhood (x, x+h) or ( x, x+ dx). Maybe we can ask Freshmeister @fresh_42 to chime in?

* Different authors have different notation.
 

FAQ: Why do we use"dx" in the derivative dy/dx?

1. Why do we use "dx" in the derivative dy/dx?

The symbol "dx" represents an infinitesimal change in the independent variable, x. In the derivative dy/dx, it signifies that we are finding the rate of change of y with respect to a small change in x. This helps us to understand how a small change in the independent variable affects the dependent variable.

2. Is "dx" just a notation or does it have a deeper meaning?

"dx" is more than just a notation, it has a fundamental meaning in calculus. It represents the concept of a limit, specifically the limit of the change in the independent variable, x, as it approaches zero. This concept is essential in understanding derivatives and integrals.

3. Can we use a different symbol instead of "dx" in the derivative notation?

Yes, in fact, there are other notations for derivatives that use different symbols. For example, Leibniz's notation uses dy/dx to represent the derivative, while Lagrange's notation uses f'(x). However, the use of "dx" is the most common notation and is often preferred due to its simplicity and clarity.

4. How does "dx" relate to the concept of the derivative?

In calculus, the derivative is defined as the limit of the change in the dependent variable divided by the change in the independent variable, as the change in the independent variable approaches zero. In other words, "dx" represents the infinitesimal change in the independent variable that is used in the calculation of the derivative.

5. Is "dx" used in all types of derivatives?

Yes, "dx" is used in all types of derivatives, including the derivatives of functions, parametric equations, and implicit functions. In each case, "dx" represents the infinitesimal change in the independent variable, which is necessary in the calculation of the derivative.

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