U1S5V03 Introducing Leibniz Notation.txt

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So far, we've been using prime notation
to write derivatives of functions.
That is, if we have some function f of x,
we write the derivative as f prime of x.
This notation was developed by Newton,
one of the fathers of calculus.
That's right, that Newton.
And this notation has some particularly nice features.
It's really compact, it's really easy to write,
and it's really easy to evaluate at specific x values.
For example, if I want to look at the derivative
at x equals 3, I simply write f prime of 3.
However, calculus was simultaneously
developed by a guy named Leibniz,
and he developed his own notation.
And this notation is still very widely used,
so it's very important that you get
used to using both notations for derivatives.
To motivate it a little bit, I want
to remind you that when we were calculating the derivative,
one way that we could do this was using the secant line
approximation.
So to find the derivative at a point
x naught we compute the slope of the secant line
some small distance delta x away.
And the slope of this line is given by delta y over delta x.
So the derivative is the limit of delta y over delta
x as delta x approaches 0.
In Leibniz's notation, we write this as dy/dx.
And I thought this was really clever
when I was in high school learning
calculus for the first time because Leibniz
was hiding the definition of the derivative inside
of the notation.
And so this was really cool to me.
I never had to memorize the definition of the derivative.
I just had to remember this notation.
And so basically what's happening
is that the small d here, this is the limit of the delta.
So the delta, remember, stands for some small difference,
and the d is the infinitesimal version.
It's what happens in the limit as it approaches 0.
Of course, this is one form of Leibniz's notation.
I don't know if you remember this,
but we actually said that we could write delta y as delta f.
So you might also see this derivative written as df/dx.
The other thing that can happen is that sometimes people
actually write the derivative as thinking about it as something
that happens to a function, so they write it
as d/dx of y or d/dx of f.
And the thing that you're going to need to keep in mind
is that all of these notations mean exactly the same thing.
So before you get started and do some practice using
this notation, I need to tell you
how you evaluate a derivative written in Leibniz's notation
at a particular x value.
The way you do this is to write an evaluation bar, like so.
So the derivative of f with respect to x at x equals 3
is written like this.
We'll be back to explain just why this notation
can be so very useful.
Good luck.
See you soon.