Okay. So, you can skip this part, but I just want to show you quickly how you would go about finding the maximum of this impedance matching factor. So, what I have to do is take the derivative of this with respect to RL and set it equal to 0, and then solve for that value RL that makes the derivative 0. So, I'm going to take the derivative of this. It's, it's a product. so, the, the derivative of this expression is going to be. So, this is RL times 1 over R internal plus RL, squared. So, this is the first term. And here's the second term. And I'm going to use the product rule. The product rule says, the derivative of this product is the derivative of the first thing times the second plus the first times the derivative of the second. So, you in turn take the derivative of one of them, multiply by the other one and then vice versa. And add it all up. So, if I do that, I take the derivative of the first with respect to RL. That's a 1. And then the second term is just R internal plus RL squared. Then, I have the second term, the derivative of that times the first term. Now, the derivative of this with respect to RL is minus 2 the power. And then it's R internal plus RL cubed. And I have to set that all equal to 0. Now, I'm going to take. To simplify this I'm going to multiply this whole equation by R internal plus RL to the third power. And that'll get rid of this factor in the denominator. And so, two of those will cancel two of those. And so, I'm going to have the first term is R internal plus RL. That's what's left. Minus 2 RL equals 0. And so, this is R internal minus RL equals zero, or RL equals R internal. Where the slope of this becomes zero, So, that's where the optimum is going to be. It's going to, it's a maximum in, in this case. Now, plugging this back in to that expression, everywhere I had RL I'll put, put in R internal. And, so I have R internal plus another R internal, that whole thing squared. And so, this whole factor becomes, this is 2R internal squared. And so, it's 1 over 4R internal squared. So the whole factor becomes this 1 over 4R internal. That. And So, the power, maximum power transfer then is this V squared times 1 over 4R internal. So, we proved that you get maximum power transfer when the load equals R internal and the max, the amount of that power that's transferred is just v squared over 4R internal. So, this is the maximum power available from a voltage source within an internal resistance are internal. [SOUND]