Question about the quotient rule of derivatives

In summary: The reason why the g(x) is squared in the denominator is because it becomes the derivative of the function with respect to x. The derivative of a function with respect to x is given by:$$\frac{d}{dx}(f(x))$$So, by squaring the g(x) in the denominator, we are getting the derivative with respect to x squared.Yes, if you know how to apply the chain rule to differentiate ##(g(x))^{-1}##.
  • #1
EchoRush
9
1
TL;DR Summary
A quick question about the theory behind the quotient rule?
Now, I understand how to use the quotient rule for derivatives and everything. I do not struggle with using it, my question is mostly about the formula itself...I very much enjoy WHY we do things in math, not just “here’s the formula, do it”...Here is the formula for the quotient rule of derivatives.

A688273F-7DB7-447F-9B43-B63AADA0DA65.jpeg

Now, my question is. Why do we square the g(x) in the denominator? I almost feel like the formula for quotient rule should just be what is in the numerator? Why do we square the g(x)? Where does that come from? Why g(x) squared?
 
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  • #2
Do you know the product rule and the chain rule? Why don't you try deriving the quotient rule using the product rule and the chain rule?
 
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  • #3
phyzguy said:
Do you know the product rule and the chain rule? Why don't you try deriving the quotient rule using the product rule and the chain rule?

would that explain why the g(x) is squared? It just seems weird to me. Why not just have g(x) function in the denom.?The fact that it is squared makes me wonder it’s origin.
 
  • #4
EchoRush said:
would that explain why the g(x) is squared? It just seems weird to me. Why not just have g(x) function in the denom.?The fact that it is squared makes me wonder it’s origin.
It is squared because ##x^n## differentiates to ## \sim x^{n-1}## and with ##n=-1## you get the one square on the right. You haven't answered @phyzguy 's question!
 
  • #5
Maybe its not a good idea to spoon feed you but sometimes there is no other way, here it is, read this Wikipedia article with 3 different proofs of the quotient rule that will help you understand the "inner mechanisms" and the ultimate why's, pick the one you like.
https://en.wikipedia.org/wiki/Quotient_rule
 
  • #6
I always have trouble remembering the order of the terms in the numerator, so I tend to use the product rule.

As people have been suggesting, try using the product rule on ##f(x) [g(x)]^{-1}## and you should see exactly where all of the terms come from. Don't take our word for it, it will really help your understanding.
 
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  • #7
RPinPA said:
I always have trouble remembering the order of the terms in the numerator, so I tend to use the product rule.
And I always thought I was the only one ...
 
  • #8
fresh_42 said:
And I always thought I was the only one ...
Me too! I realized early on that the quotient rule was just a consequence of the product rule, so I didn't need to memorize the quotient rule. I never use it.
 
  • #9
RPinPA said:
I always have trouble remembering the order of the terms in the numerator, so I tend to use the product rule.
fresh_42 said:
And I always thought I was the only one ...
I came up with my own mnemonic device for the quotient rule, and one I've never seen anywhere else. Here it is, in the context of the differential of u/v.
$$d(\frac u v) = \frac{v du - u dv}{v^2}$$
How do I remember which term in the numerator comes first? The vd one, an abbreviation for something unrelated to mathematics.

For a derivative instead of a differential, replace du by du/dx or u' and similar for dv.
 
  • #10
The derivative of [itex]\frac{f(x)}{g(x)}[/itex] is given, using the definition of the derivative, by [itex]\lim_{h\to 0}\frac{\frac{f(x+h)}{g(x+h)}- \frac{f(x)}{g(x)}}{h}[/itex].
To do that quotient, [itex]\frac{f(x+h)}{g(x+h)}- \frac{f(x)}{g(x)}[/itex], get the "common denominator", [itex]g(x)g(x+ h)[/itex]: [itex]\frac{f(x+ h)g(x)}{g(x)g(x+h)}- \frac{f(x)g(x+h)}{g(x)g(x+h)}[/itex]. (it is that "g(x)g(x+h)" in the denominator that will give "[itex]g^2(x)[/itex]" after we take the limit.)
 
  • #11
Mark44 said:
I came up with my own mnemonic device for the quotient rule, and one I've never seen anywhere else. Here it is, in the context of the differential of u/v.
$$d(\frac u v) = \frac{v du - u dv}{v^2}$$
How do I remember which term in the numerator comes first? The vd one, an abbreviation for something unrelated to mathematics.

For a derivative instead of a differential, replace du by du/dx or u' and similar for dv.
Another way is like this:
Function (Hi)/(Ho)

Mnemonic: Ho d Hi minus Hi d Ho, over Ho Ho

Symbolified: (Ho*dHi-Hi*dHo)/(HoHo)
(Try writing on paper using better 'typesetting' to see it better.)
 
  • #12
EchoRush said:
would that explain why the g(x) is squared?

Yes, if you know how to apply the chain rule to differentiate ##(g(x))^{-1}##.

I find the explanation given by @HallsofIvy the most intuitive explanation of the denominator in the quotient rule. To explain the numerator, you need the same trick used to prove the product rule.
 
  • #13
The Quotient Rule Rhyme (for D hi/low):
If the quotient rule you wish to know,
It's low Dhi less hi Dlow
Draw a line and down below
The denominator squared must go.
 
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1. What is the quotient rule of derivatives?

The quotient rule is a mathematical formula used to find the derivative of a quotient of two functions. It states that the derivative of f(x)/g(x) is equal to (g(x)*f'(x) - f(x)*g'(x)) / (g(x))^2.

2. When is the quotient rule used?

The quotient rule is used when finding the derivative of a function that is expressed as a quotient of two other functions. It is particularly useful when the quotient cannot be simplified using algebraic manipulation.

3. How do you apply the quotient rule?

To apply the quotient rule, you first identify the two functions in the quotient. Then, you take the derivative of each function separately and plug them into the formula: (g(x)*f'(x) - f(x)*g'(x)) / (g(x))^2. Simplify the resulting expression to get the derivative of the original function.

4. Can the quotient rule be combined with other derivative rules?

Yes, the quotient rule can be combined with other derivative rules such as the power rule and the chain rule. This is useful when finding the derivative of a function that is a combination of multiple functions.

5. What are some common mistakes when using the quotient rule?

Some common mistakes when using the quotient rule include forgetting to square the denominator, mixing up the order of the functions in the numerator, and not simplifying the resulting expression. It is important to double check your work and simplify the final answer to avoid these mistakes.

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