U1S6V04 Introducing higher derivatives.txt

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We started with this function f and we asked you
to take the derivative of f.
And you should have gotten 3x squared minus 6.
So that's f prime of x.
And since it's a function itself, we called it g of x.
But now we can find g prime of x.
That's just 6x.
So this g prime is the derivative
of the derivative of f.
And we call it the second derivative of f.
The rest of this sequence is devoted
to what this second derivative means, why it's important.
But for now, we'll just focus on the notation.
How do we even write this?

Well, the derivative of f, which we
can call the first derivative of f, if we need to be explicit,
that's just f prime.
And the second derivative of f is the derivative
of f prime, so its f prime prime.
And we generally write that as f double prime.
Great.
Now, let me clear some space here.
We've got f prime of x as the first derivative
and f double prime of x as the second derivative.
That's prime notation, but what about Leibniz notation?
In Leibniz notation, the first derivative of f is df/dx.

And one way to interpret this is when we differentiate
f with respect to x, we're doing or applying this operation
of d/dx to the function f.
That means that when we take a derivative of this,
we apply d/dx again.
So it's going to be d/dx of d/dx of f.
And it's the derivative with respect
to x of the derivative with respect to x of f.
And we can write that more compactly
as quantity d/dx squared of f.
We're just doing this thing twice.

And then most people expand this square out
and they'll write d squared over dx squared f.
So it takes a little bit of getting used to.

Of course, if we wanted to, we could keep going
and we could take the derivative of the second derivative
to get the third derivative.
And that would be written d cubed over dx
cubed f in Leibniz notation.
Or in prime notation, f prime prime prime of x.
Sometimes, if there are too many primes, you might write f
and then parentheses 3 in the superscript of x.
But the most important are the first
and the second derivatives, so that's
what we're going to focus on most in this course.
To start, we have a series of questions
to help you figure out just why and how this second derivative
is meaningful geometrically.
So enjoy.