Problems with signs . . . . . .?

(1) (x - y)^2 = (y - x)^2.....Multiply them out to be sure of this yourself.

(2) However (x - y)^3 ≠ (y - x)^3.....Multiply them out - same reason.

(3) x - y ≠ y - x

1 / (x - y) = 1 * (x - y)^(-1) = (x - y)^(-1)

When you differentiate in terms of y, the internal derivative, that of (x - y) = (dx/dy - 1).

However you are not considering the dx/dy so the internal derivative is (-1)

You must ***never forget the internal derivative***. However the last time that I taught calculus was 36 years ago and it may no longer be called the internal derivative...But it is the derivative of the (x - y) in the expression (x - y)^(-1). This is actually an application of the chain rule of differentiation.

So d(1 / (x - y)) = the internal derivative (-1) * the exponent (-1) * (x - y)^2

From (1) we know that this is (-1) * (-1) * (y - x)^(-2) =( (y - x)^(-2)) (because minus a minus is plus) and that is equal to:

(1 / (y - x)^2)

I have no idea why they changed (x - y)^2 to (y - x)^2. While it is true that they are equal, I can't see any reason to do it and I think that it would be very confusing for most students. It's totally pointless and counterproductive.

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Now for the second derivative of the original function which is simply the derivative of the first derivative of that function which is:

(1 / (y - x)^2) = (y - x)^(-2)

This time the internal derivative is +1 and thus of no consequence.

So we have the -2 (from the old exponent) * (y - x)^(-3) because the new exponent is equal to the old exponent - 1 (and -2 - 1 = -3).

The answer is (-2) * ((y - x)^(-3)) = (-2) / (y - x)^3....<<<..Answer to 2nd derivative.

The reason that the signs don't change back is because they are now working with (y - x)^3 which is equal to minus (x - y)^3 (multiply them out so that you are sure of this.)

Since (y - x)^3 = -(x - y)^3, then

(-2) / (y - x)^3 = (-2) / -(x - y)^3 and since minus divided by minus is plus:

The answer is equivalent to (2 / (x - y)^3) and the signs really do change back. I told you that it was pointless and confusing for them to change (x - y)^2 to (y - x)^2.

That is where you got lost and what student wouldn't get lost with this stupid change?

I may sound opinionated but that change was the source of your confusion and I can't imagine why they did it.

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