Is the Quotient Rule Necessary?

[Superstring]

If you can write, for example
$$\frac{cosx}{x+1} = (cosx)(x+1)^{-1}$$
then what is the point of the quotient rule? Can't you just use the product rule for everything?

 

[Hurkyl]

If you can write, for example
$$\frac{cosx}{x+1} = (cosx)(x+1)^{-1}$$
then what is the point of the quotient rule? Can't you just use the product rule for everything?

Yep. (don't forget you're using the power rule and chain rule too )

But the quotient rule comes up often enough it's more convenient to use it directly.

 

[LCKurtz]

If you can write, for example
$$\frac{cosx}{x+1} = (cosx)(x+1)^{-1}$$
then what is the point of the quotient rule? Can't you just use the product rule for everything?

Yes. But using the product rule for such as that frequently requires an extra simplification step to combine the fractions. And for many students, such manipulations as adding fractions with algebra expressions seem to be more difficult than the calculus. With the quotient rule you automatically get a single fraction.

 

[hgfalling]

Also if you think of it as an extension of the product rule, you'll never forget the order of the terms in the numerator of the quotient rule, because the minus sign comes from taking the derivative of the denominator $g(x)^{-1}$.