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Derivative notation review

Review the different common ways of writing derivatives.
Lagrange's notation: f, prime
Leibniz's notation: start fraction, d, y, divided by, d, x, end fraction
Newton's notation: y, with, \dot, on top

What is derivative notation?

Derivatives are the result of performing a differentiation process upon a function or an expression. Derivative notation is the way we express derivatives mathematically. This is in contrast to natural language where we can simply say "the derivative of...".

Lagrange's notation

In Lagrange's notation, the derivative of f is expressed as f, prime (pronounced "f prime" ).
This notation is probably the most common when dealing with functions with a single variable.
If, instead of a function, we have an equation like y, equals, f, left parenthesis, x, right parenthesis, we can also write y, prime to represent the derivative. This, however, is less common to do.

Leibniz's notation

In Leibniz's notation, the derivative of f is expressed as start fraction, d, divided by, d, x, end fraction, f, left parenthesis, x, right parenthesis. When we have an equation y, equals, f, left parenthesis, x, right parenthesis we can express the derivative as start fraction, d, y, divided by, d, x, end fraction.
Here, start fraction, d, divided by, d, x, end fraction serves as an operator that indicates a differentiation with respect to x. This notation also allows us to directly express the derivative of an expression without using a function or a dependent variable. For example, the derivative of x, squared can be expressed as start fraction, d, divided by, d, x, end fraction, left parenthesis, x, squared, right parenthesis.
This notation, while less comfortable than Lagrange's notation, becomes very useful when dealing with integral calculus, differential equations, and multivariable calculus.

Newton's notation

In Newton's notation, the derivative of f is expressed as f, with, \dot, on top and the derivative of y, equals, f, left parenthesis, x, right parenthesis is expressed as y, with, \dot, on top.
This notation is mostly common in Physics and other sciences where calculus is applied in a real-world context.

Check your understanding

Problem 1
  • Current
g, left parenthesis, x, right parenthesis, equals, square root of, x, end square root
How can we express the derivative of square root of, x, end square root?
Choose all answers that apply:

Want to join the conversation?

  • leaf blue style avatar for user Tenacious
    Why haven't I seen Newton's method - the 'dot' - in any of my college calc courses?
    (115 votes)
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    • blobby green style avatar for user Christopher Clements
      I have used dot notation to a great extent in classical mechanics to note first(dot) and Second(double dot) derivatives...more out of laziness than anything else. "Pure" mathematicians rarely, if ever, use this notation in my experience. Many are in fact rather critical of this notation in all cases save 4th or greater derivatives where even the theorists get lazy :)
      (40 votes)
  • starky sapling style avatar for user brycepietila
    Just curious. Why is dy/dx a correct way to notate the derivative of cosine or any specific function for that matter? If I only wrote dy/dx on a piece of paper and asked somebody to differentiate, then I would hope they would not say that the derivative is negative sine
    (52 votes)
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  • leaf blue style avatar for user octmacs
    Leibniz's notation made me confused a lot when I first met it in calculus integral. I always thought it is kind of y/x rather than y', for I had already seen dx in integral a long time ago before I seen dy/dx. It seems that this notation is far different from the other two, for does it have other functions when written differently?
    (15 votes)
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  • leaf green style avatar for user KHJ
    I heard that newton's notations are very complex as it uses many notations like dots or hats in random. Is that true?
    (11 votes)
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    • piceratops ultimate style avatar for user Scottyboy1
      It's not that it's more complex, it's just that he was a physics and math wiz. His dots can add up quickly in mathematics because you might be taking the 10th (for example) derivite of some quantity. That would look like this:

      .
      .
      .
      .
      .
      .
      .
      .
      .
      y

      No thanks! Lol


      In physics you are just looking for the first derivative (velocity) and the second derivate (acceleration).........and once in a while (like almost never) you will need the third derivative, which is (jerk).

      So the most dots you would get are 3.

      When you think about it, d/dx is a lot to write down when you can just write (dot). But too many dots would make each equation too cumbersome to write.
      (42 votes)
  • marcimus orange style avatar for user Harsh Chauhan
    What is difference between derivative and differentiation?
    (10 votes)
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  • orange juice squid orange style avatar for user Clara Wessels
    In Leibniz notation, when would you use dy/dx and when would you use d/dx? Or are these two notations interchangeable?

    Am I correct to say that when the function is given in the form f(x)=..., you would write the derivative as d/dx(fx), and when given as y=..., you would write it as dy/dx(y)?
    (15 votes)
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    • blobby green style avatar for user Anuvesh Kumar
      You can think of Leibniz notation as just d/dx of something which means it's the derivative of something w.r.t. x.

      1. If that something is just an expression you can write d(expression)/dx.

      so if expression is x^2 then it's derivative is represented as d(x^2)/dx.

      2. If we decide to use the functional notation, viz. f(x) then derivative is represented as d f(x)/dx.

      Note that 'f(x)' is not a variable, all it says is that f is a function of x, which is given by some

      You can imagine if the expression is large, it'll be cumbersome to write d(expression)/dx. So this notation as well as the next is useful in that context.

      3. If instead of using functional notation we decide to use the notation of dependent variable, as in the value of the variable depends on something, where the something can be either an expression or a function.
      so y = x^2 or y = f(x)

      then dy/dx then represents the derivative of dependent variable w.r.t x.

      TO SUMMARIZE:

      dependent variable = function of x = expression

      OR y = f(x) = x^2

      then,

      dy/dx = d(f(x))/dx = d(x^2)/dx
      (5 votes)
  • blobby blue style avatar for user User 5939293
    Is this correct for y=cos(x):
    d cos(x) / dx
    (10 votes)
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  • mr pants purple style avatar for user lj08197
    I'm middle school i dont get the difference between d/dy and dx/dy and dy/dx etc.
    (4 votes)
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    • hopper cool style avatar for user Iron Programming
      Howdy lj08197,

      What you are asking about is called the Leibniz notation for derivatives. With this notation, d/dx is considered the derivative operator. So if we say d/dx[f(x)] we would be taking the derivative of f(x). The result of such a derivative operation would be a derivative. In our case, we took the derivative of a function (f(x), which can be thought as the dependent variable, y), with respect to x. We write that as dy/dx.

      Let's look at some examples.
      (1.) d/dx[f(x)] = dy/dx (we took the derivative of f(x) with respect to x)
      (2.) d/dt[f(t)] = dy/dt (we took the derivative of f(t) with respect to t)
      (3.) d/dt[f(x)] = Not Application (N/A). There is no variable t in this function!
      (4.) d/dx[2x + 3] = Take the derivative of the expression "2x + 3" with respect to x. You will learn how to do this later.

      If you are comparing this notation to other notation, such as f' (pronounced f prime), then dy/dx would be the equivalent of f'(x), the derivative of f(x).

      Hope this helps!
      (16 votes)
  • old spice man green style avatar for user bhannon039
    In my physics book, it appears there is some sort of algebra involved using the derivative notation itself. For example, with the work-kinetic energy theorem there is the following result:

    dv/dt = (dv/dx)*(dx/dt) = (dv/dx)*v

    It wasn't explained in the book and I am trying to find where I can figure out where this came from.
    (5 votes)
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    • mr pink red style avatar for user andrewp18
      That is the chain rule in action!
      [𝑓(𝑔(𝑥))]' = 𝑓'(𝑔(𝑥))𝑔'(𝑥)
      In Leibniz notation this is:
      (𝑑𝑓)/(𝑑𝑥) = [(𝑑𝑓)/(𝑑𝑔)] • [(𝑑𝑔)/(𝑑𝑥)]
      In your case, (𝑑𝑥)/(𝑑𝑡) is velocity 𝐯 (change in position per change in time). I suggest you watch the videos on Chain Rule. Comment if you have questions!
      (12 votes)
  • boggle blue style avatar for user lupinx2
    Why is
    d
    -- g(x)
    dx
    a correct notation for the derivative of g(x)? shouldn't it be dy on top?
    (3 votes)
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