Partial Derivatives. Why and when to avoid the quotient rule?

[504aldo]

Hello PH,

This is my first post. I came here while studying partial derivatives and after clicking here and there for over 4hrs for an answer. While practicing the derivatives rules i came across the hideous quotient rule. I've solved around 20 fractional problems trying to find a decision tree that will help me understand why and when to use (or not to use) the quotient rule.

I have no problem in rewriting the function to another one with a negative exponent in the numerator and use the product/chain/power rule when necessary.

So, is it possible to precisely know when to avoid the quotient rule?

 

[Dick]

Hello PH,

This is my first post. I came here while studying partial derivatives and after clicking here and there for over 4hrs for an answer. While practicing the derivatives rules i came across the hideous quotient rule. I've solved around 20 fractional problems trying to find a decision tree that will help me understand why and when to use (or not to use) the quotient rule.

I have no problem in rewriting the function to another one with a negative exponent in the numerator and use the product/chain/power rule when necessary.

So, is it possible to precisely know when to avoid the quotient rule?

You never HAVE to use the quotient rule. You can always write f(x)/g(x) as f(x)*g(x)^(-1) and use the product and chain rule. It is completely up to you. There's no decision tree. it's a preference tree. And you have to fill in what your preference is.

 

[LCKurtz]

I would add that one advantage of using the quotient rule for a fraction is that the answer comes as a single fraction. If you use the product rule, you have to add the fractions if you wish to simplify it, which is frequently the case. The exception is if the numerator is constant so it really isn't a fractional function of $x$.