00:01
This problem wants us to prove that the derivative of y equals 4x squared minus 5 divided by 3 plus 2x when using the quotient rule is equal to the derivative when using the product rule.
00:11
So first we will look at the derivative using the quotient rule where we set up our derivative as the low function 3 plus 2x times the derivative of the high function, which in our case would be the derivative of 4x squared minus 5, which would be 8x.
00:24
And that's low derivative of high, and then we subtract the high function times the derivative of the low, and the derivative of 3 plus 2x would be 2, and that's all over low squared of 3 plus 2x squared, and now we'll leave that in the form that it is currently because that does represent the derivative, even though we could clean it up a little bit, but to prove that these will give us the same derivative no matter what, but we will now look to use the product rule.
00:56
And to use the product rule, we're going to change the form of the original function to make this a product because we can take 3 plus 2x to the top of our fraction and show that as 4x squared minus 5 times 3x, or excuse me, 3 plus 2x raised to the negative first.
01:14
That way we can bring it to the top of our fraction.
01:17
And now we can use the product rule where we look at the derivative of the first set, which would be 8x, and no derivative of the second set, so that would be 3 plus 2x raised to the negative first, and then plus the other combination where we don't take the derivative of the first set, and then we take the derivative of the second set where we'd have to use chain rule, because negative 1 would have to come in front and multiply by our parentheses, so that's multiplying by negative 1 times 3 plus 2x, and then we take 1 away from the exponent to make that negative second, and that's times the derivative of the input for our chain rule, which would be 2...