The last of the rules which will help us differentiate a variety of functions is the quotient rule. It allows us to differentiate quotients:

Here it is:

I remember this with the silly rhyme:

Low d high, minus high d low, draw the line and square below

where low d high means $g(x)f^{\prime }(x)$ and high d low means $f(x)g^{\prime }(x).$

As before, we'll get used to using this rule before proving it.

Differentiate

1. $$y=\frac{x^2}{x^2-1}$$
2. $$y=\tan (x)$$
3. $$\frac{\cos (2x)}{x{e}^x}$$