[music]. Let's combine the change rule and the
derivative of sign to differentiate a slightly more complicated function.
Let's try to differentiate sine of x squared.
We can realize this function as the composition of two functions.
I can write sine of x squared as the composition of f and g, where f is the
sine function and g is the squaring function.
Now, how do I differentiate a composition of two functions?
I use the chain rule. So I differentiate f and the derivative of
f is cosine x, where the derivative of sine is cosine.
And the derivative of g is just 2x. So now I want to differentiate the
composition of f and g and that's by the chain rule f prime of g of x times g prime
of x. In this case, f prime is cosine, so it's
cosine of just g of x, which is x squared times the derivative of g, which is 2 x.
So, the derivative of sine x squared with respect to x, is cosine of x squared times
2x. Honestly, this is a pretty neat example.
In magnitude, this function, sine of x squared, is no bigger than 1.
And yet, what do we know about this function's derivative?
Well, the derivative of sine of x squared is cosine of x squared times 2x.
And that function can be as large as you like.
You can make cosine of x squared times 2x as big as you want, as long as you choose
x appropriately. So what we have here is a function which
isn't very big. The function's value is no bigger than 1
in magnitude, but the function's derivative is very large.
And you can see that on the graph. The values of this function really aren't
that large. The values are all hugging zero.
But the derivative, the slope of the tangent line is enormous.
Look over here. If you imagine a tangent line, that
tangent line is going to have enormous slope.
The derivative over here is going to be very large.
In spite of the fact, that the actual values of the function really aren't that
large. This actually provides another lesson.
Just because 2 functions are nearby in value, doesn't mean that their derivatives
are anything close to each other. For instance, here is the graph of the
cosine function. And here is the graph of a function sine
of x square over 10 plus cosine of x. Since sine of x squared is between minus 1
and 1, this differs from the cosine by no more than a tenth.
And, yeah, you can see the graph is really close to the graph of cosine and yet this
graph is way more wriggly. Let's zoom in and we can see the same sort
of thing. Here's a zoomed in copy of just a cosine
curve. And here's what happens if you zoom in on
this other function. And in terms of the value, this other
function really isn't different from cosine very much.
But in terms of derivative, this function is totally different than cosine.
This function is super wiggly, so the derivative of this function is enormous,
even though the derivative of cosine is no bigger than 1 in magnitude.
If you think this is kind of an interesting example, it's worth trying to
cook up an even more elaborate example. Here's a very specific challenge for you.
Can you find a function that, just make one up, so that your functions values and
magnitude are less than c, and your functions derivative and magnitude is less
than c? So I want a specific number c, so that no
matter what value of x you plug in, the function's value there and the function's
derivative there is less than c in magnitude.
But, I want that function to have second derivative which can't be bounded by a
constant. I want you to cook up a function so that,
yeah, the values of the first derivative are bounded by c.
But the second derivative can be as big or as negative as I'd like, by choosing x
appropriately. Can you find a function like that?