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### Course: AP®︎/College Calculus BC > Unit 2

Lesson 1: Defining average and instantaneous rates of change at a point- Newton, Leibniz, and Usain Bolt
- Derivative as a concept
- Secant lines & average rate of change
- Secant lines & average rate of change
- Derivative notation review
- Derivative as slope of curve
- Derivative as slope of curve
- The derivative & tangent line equations
- The derivative & tangent line equations

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# Derivative notation review

Review the different common ways of writing derivatives.

**Lagrange's notation:**

**Leibniz's notation:**

**Newton's notation:**

## 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 $y=f(x)$ , 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 $\frac{d}{dx}}f(x)$ . When we have an equation $y=f(x)$ we can express the derivative as $\frac{dy}{dx}$ .

Here, $\frac{d}{dx}$ 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 ${x}^{2}$ can be expressed as $\frac{d}{dx}}({x}^{2})$ .

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 $\dot{f}$ and the derivative of $y=f(x)$ is expressed as $\dot{y}$ .

This notation is mostly common in Physics and other sciences where calculus is applied in a real-world context.

## Check your understanding

## Want to join the conversation?

- Why haven't I seen Newton's method - the 'dot' - in any of my college calc courses?(121 votes)
- 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 :)(44 votes)

- 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(54 votes)
- Alex, you are 100% correct. If the function is not know dy/dx simply means some derivative y in relation to x(27 votes)

- 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?(18 votes)
- yup "d" just small change ......very small we can say that """d""" is just small DELTA(5 votes)

- I heard that newton's notations are very complex as it uses many notations like dots or hats in random. Is that true?(12 votes)
- 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.(43 votes)

- What is difference between derivative and differentiation?(12 votes)
- Differentiation is the process of finding a derivative of a function, I think you could say.(35 votes)

- 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)?(19 votes)- 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(6 votes)

- Is this correct for y=cos(x):

d cos(x) / dx(11 votes)- Yes, that's correct.(6 votes)

- I'm middle school i dont get the difference between d/dy and dx/dy and dy/dx etc.(5 votes)
- 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!(17 votes)

- 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.(6 votes)- 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!(14 votes)

- Why is

d

-- g(x)

dx

a correct notation for the derivative of g(x)? shouldn't it be dy on top?(3 votes)- d/dx is an operation that means "take the derivative with respect to x" whereas dy/dx indicates that "the derivative of y was taken with respect to x".(13 votes)