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## AP®︎/College Calculus AB

### 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

AP.CALC:

CHA‑2 (EU)

, CHA‑2.B (LO)

, CHA‑2.B.2 (EK)

, CHA‑2.B.3 (EK)

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

## Want to join the conversation?

- 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(3 votes)
- Alex, you are 100% correct. If the function is not know dy/dx simply means some derivative y in relation to x(2 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?(14 votes)
- yup "d" just small change ......very small we can say that """d""" is just small DELTA(4 votes)

- 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)
- 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.(36 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)?(11 votes) - What is difference between derivative and differentiation?(9 votes)
- While studying, I had the same question and this is what I found which helped me to understand this:
*"A function is differentiable if it has a derivative, and a function has a derivative if it is differentiable."*(4 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.(5 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!(11 votes)

- Why is

d

-- g(x)

dx

a correct notation for the derivative of g(x)? shouldn't it be dy on top?(2 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".(7 votes)

- The explanation of Lagrange isn't clear because it says "the derivative of f is expressed as f' ", not as f'(x). The opening summary reinforces this notion. So in problem 1, one would think that g'(x) is
**not**a correct answer and conversely in problem 2, one could conclude that cos'**is**a correct answer, based on the previous explanations. The problem explanations correct these earlier confusing statements but it's too late (unless you think that making mistakes is a proper pedagogic strategy).(4 votes) - I'm middle school i dont get the difference between d/dy and dx/dy and dy/dx etc.(2 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!(3 votes)

- Can someone explain to me why 'cos' is considered a "mathematical operator" and not a function? Because syntactically it looks like a function to me, and it also seems to match the formal definition of a function.(2 votes)
- a function needs an input and an output. so cos(x) = y is a function that uses the operator cos. this is the same for anything like +,-,*, roots, exponents, logs and everything else.(2 votes)