And we figured that out from the derivative.

From here to here the derivative is less than 0.

So, we get 4i plus... and then the derivative the second derivative, or the second derivative of y with respect to t is just the derivative of the first derivative.

So our second derivative is the derivative of this.

Well, the highest derivative here is the second derivative.

So the derivative of c with respect to x, just from the chain rule, is equal to the derivative of c with respect to y times the derivative of y with respect to x.

1st derivative

2nd derivative

The Derivative.

So y of 0 is kind of one derivative away from that.

Derivative of this is equal to the derivative of this.

You could take the derivative, and the derivative is continuous.

And this, of course, is just going to be equal to the derivative of x of the derivative of x the second derivative of x(t) is just the derivative of the first derivative.

Well, the second derivative is just the slope of the derivative.

So the derivative of the numerator is the derivative of 4x squared is 8x minus 5 over the derivative of the denominator is, well, derivative of 1 is 0.

Show first derivative

Show 1st derivative

Show second derivative

Show 2nd derivative

Derivative chain rule.

Derivative equals 0.

That's the derivative.

The second derivative.

Or the derivative.

1st derivative r.

This is the anti derivative and this is also the anti derivative.

So the Laplace Transform of y prime prime, if we apply that, that's equal to s times the Laplace Transform of well if we go from y prime to y, you're just taking the anti derivative, so if you're taking the anti derivative of y, of the second derivative, we just end up with the first derivative minus the first derivative at 0.

Remember, all of this, this was all just from my definition of a derivative.

Take the derivative operator because we're taking the derivative with respect to x.

Well, you could view it as the derivative of the first derivative, right?

I'll write the derivative operator, the derivative oh y with respect to x.

Let's take the derivative of that, put it over the derivative of that.

Or the second derivative.

What's the second derivative?

So what's it's derivative?

lot of derivative problems.

2nd derivative r squared.

Well what's the derivative?

So this is derivative.

They're the points where either the derivative is 0, or the derivative doesn't exist.

This is the derivative of this, just like this is the derivative of this.

Because we took the derivative, and clearly the derivative of x squared is 2x.

But what's the derivative of x squared what's the derivative x squared plus 1?

Well they're saying that the second derivative is greater than 0 from 0 to 5.

Take the derivative of the first term and the derivative of the first term isn't a joke you take the derivative of the inside first, let's do the chain rule, derivative of the inside first.