Derivatives

Well, just do the chain rule.

And the answer is C.

Let's start with the first expression. This one over here doesn't depend on sigma at all.

Here, when you take the subtraction, you're taking the one that had a derivative, now it doesn't.

Well, we could just keep doing this, and actually, we will keep doing this, and you know, just saying, well, the zeroth derivative, or at the value, is the same the first derivative is the same at 0, the second derivative is the same at 0, we'll say the third derivative, the fourth derivative, and we'll keep doing that.

And of course these aren't functions.

Our solution has e in it.

And the derivative of the inside is minus 3.

And so you could say well, 2X is the derivative of X 2.

So if we have u times v, if we take the derivative with respect to t of that, that's equal to the derivative of the first times the second function plus the first function times the derivative of the second function.

And what's the third derivative? f 3 of x.

1st derivative

2nd derivative

Derivative of the inside is r.

What's y1 prime?

Let's see if we can find a relationship between the

And where does the c come from?

2nd derivative r squared.

Well, the third derivative is going to be, so p prime prime prime of x, we could have also written p3 of x, is equal to the derivative of this.

But if you want to do these problems really quick, you just substitute the 2nd derivatives with an r squared, the first derivatives with an r, and then the function with oh sorry, no.

So this was our original problem.

y prime prime.

Or you could have said divided by this, either way.

You could take the sixth derivative.

I could have written u of x times u of x.

The first derivative of p of x, this is my current p of x, this constant term, derivative of 0.

Because we evaluated this curvy function, it's second derivative 0, so we just got a number here.

So the second derivative of y2 is just e to the x plus 2 times the first derivative is what?

So y soon. prime is equal to what?

What's the first derivative of p?

This is our current p.

Well, what's the derivative of t?

And now, let's just extend this further.

And then you take the derivative of the inside.

We have a negative sine here, so it becomes a minus one.

It's going to be minus x to the sixth over 6 factorial.

I think you see what I did.

So I can just rewrite that as A so now A is not a function anymore, it's just a number A times the second derivative of y, with respect to x, plus B times the first derivative, plus C times y.

What is a differential equation?

Optimizing the log is the same as optimizing p because the log is monotonic to p.

And now, let's just do a little bit of take a step back and think about what kind of function if I have the function and I take its derivative and then I take its second derivative, most times I get something completely different.

Show first derivative

Show second derivative

Derivative chain rule.