When you’re dealing with an expression like

it is often useful to do to it the algebraic equivalent of rewriting as .

Only, what you do here is rewrite the expression in two parts, one of which is just a constant, and the other of which has no “x” in the numerator.

For example

Now you can differentiate the expression without having to use the quotient rule.

You can integrate it without having to worry about whether you should look for a substitution.

You can sketch it easily. The example is a rectangular hyperbola with one asymptote at y = 1 and another at x = −1, but “upside down” (the bit above the y-asymptote is on the left, not the right).

In general you get a rectangular hyperbola with one asymptote at and the other at . Depending on the sign of bc−ad, the hyperbola may or may not be “upside down”.

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