Why should one use the quotient rule instead of the power rule to differentiate a quotient?

Question

There's a lot of emphasis on the difference quotient, as it's on the AP test and all that, but honestly using the power rule plus the product rule is soo much easier and gives you the same answer and slope. What am I missing here?

For example, if you wanted to differentiate $$f(x)=\frac{x^2+4x-2}{x-1},$$ you could either do $$f'(x)=\frac{d}{dx}[(x^2+4x-2)(x-1)^{-1}]=(2x+4)(x-1)^{-1}-(x^2+4x-2)(x-1)^{-2}, $$ or you could do $$ f'(x)=\frac{(x-1)(2x+4)-(x^2+4x-2)(1)}{(x-1)^2}. $$

Answer 1

I'm confused by the question. In AP Calc classes you need to know both, but usually on free-response they will let you derive equations using method of choice. You might need to be able to recognize a difference quotient in a multiple choice or free-response, and it is important to understand the intuition behind it! However, you surely don't have to solve every problem with it. Hope this helps!

Answer 2

There are two reasons why the quotient rule can be superior to the power rule plus product rule in differentiating a quotient:

1. It preserves common denominators when simplifying the result. If you use the power rule plus the product rule, you often must find a common denominator to simplify the result. The tradeoff here is between more calculus/less algebra, and less calculus/more algebra. As I find the calculus to be less error-prone than the algebra, I would opt for the more calculus/less algebra approach.

2. There are a few, a very few, integrals that succumb to the quotient rule (in reverse). If you are very familiar with the quotient rule, you can sometimes spot these integrals.

If you find the quotient rule difficult to remember, here's a mnemonic device that can help you out: "low, dee-high minus high, dee-low over the square of what's below". There are countless variations on this mnemonic device, but I think the quotient rule is worth keeping in your arsenal, despite being, technically, unnecessary. You should have it for the same reason you should have the quadratic formula memorized, even though, if you know how to complete the square, you don't need the quadratic formula: it's faster and less error-prone.

Answer 3

Maybe if you go up to the AP test, and then never do any math after that, it will not matter. But is that what you plan?

For example, when you work on differential equations, it is often helpful to recognize an expression of the form $u v' - u' v$ as related to the derivative of a quotient.