The Derivative.

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We take the antiderivative.

So the derivative of one is just zero, the derivative of negative 2 sine of x...

So the instantaneous slope is the derivative.

What's the derivative of this?

And what's the derivative of this?

What does Tony stand for?

What's the derivative of so we can have 100 constant out there times the derivative of this inside function.

Sorry, we have that derivative 1.

And then the second derivitive

And if I wanted to take the partial of this, with respect to no, I want to take the derivative, not the partial the derivative of this, with respect to x, this is equal to the partial of psi, with respect to x, plus the partial of psi, with respect to y, times dy, dx.

I'm going to add minus 3 times the first derivative to that.

What's the derivative of cosine of x?

Well, the derivative of e to the x is e to the x.

Plus the derivative of this with respect to why is 3 times we're just doing the chain rule the derivative of y with respect to x.

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

Plus the derivative of this with respect to x.

The derivative of a vector valued function is once again going to be a derivative.

If you don't believe me, take the derivative of this.

The female version of the name is Stephanie.

Or if we use calculus terminology we would view it as the derivative.

Well, the derivative of well, let me make it even simpler.

What is the derivative of psi with respect to x?

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

The second derivative is equal to what?

So the second derivative is that.

1 2 sine of 2 tau is the derivative of this.

The derivative of psi with respect to x is equal to 0.

And then you take the partial derivative of M, with respect to y, and then you take the partial derivative of N, with respect to x, and they are equal to each other, then and it's actually if and only if, so it goes both ways this is an exact equation, an exact differential equation.

We had the minus out here, but then we had a minus and we took the antiderivative and they canceled out, so we have k q1 q2 times the antiderivative evaluated at 5, so 1 over 5, right? r to the negative 1, so 1 over r minus the antiderivative evaluated at 10 minus 1 over 10 is equal to well, 1 over 5, that's the same thing as 2 over 10, right?

The second derivative of y minus 3 times the first derivative y minus 4 times the function.

Because this is the derivative of psi with respect to x.

So the derivative of distance is velocity, the antiderivative of velocity is distance.

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.

The second derivative of y minus 3 times the first derivative minus 4 times y is equal to and now instead of having an exponential function or a trigonometric functional, we'll just have a simple well, it just looks an x squared term, but it's a polynomial.

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

But we're evaluating the derivative of that function now of 0.

Plus the derivative of this with respect to x is easy, 2.

So the first expression, you take the derivative of that with respect to x.

And in the last video we thought about, well, what does it mean to take the derivative of a vector valued function?

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But let's just figure out what that derivative is for all of the other values for x.