Differentiation: Quotient Rule

The basic formula for the quotient rule is.

f(x) = (g(x)/h(x))

So

f’(x) = [(g'(x).h(x)) - (g(x).h'(x))] / h(x)^2

This may be tricky to see as I’m typing it out but if you look at it long enough you’ll understand it.

A break down of this formula. We want f’(x) and in this state we are given f(x) = g(x)/h(x). So deriving that would possibly be hard and take a long time without the quotient rule. So on the right hand side for the differentiated formula we see the product of differentiated g(x) [g'(x)] multiplied by h(x) then g(x) multiplied by the derived h(x) [h'(x)] the right term is subtracted off the left term and then both are divided by h(x)^2. Understand now?

A better way of understanding these rules is putting them into motion, so I’m going to put down a few questions and you’re going to try answer them. I will work through them afterwards and show any interesting points we should acknowledge.

1. y = (x^2)/(2x) (simple one here for you)
2. y = (x+1)/(x^2) (getting a bit stranger, but it’s cool, it’s cool)
3. y = (x^2+3x)/(x+1)
4. y = ((x+1)^2)/(2x)
5. y = ((x+1)^2)/((x-1)^2)

These are all very easy and there is no doubt you will see harder questions if you’re taking a calculus course, so lets break them down and show you what to do.

1. [((2x).(2x)) - ((x^2).(2))] / (4x^2) here we a very basic question, simplifying in this case can be done but probably doesn’t need to be as a flip side to using the quotient rule we can use normal differentiation as the above question can be simplified. As such x/2, deriving this would give you 1/2 so to prove the quotient rule I’ll simplify the answer we got by using the quotient rule. (4x^2 – 2x^2)/(4x^2) this gives 2x^2 in the numerator and 4x^2 in the dominator which is simplified to 1/2. This proves the quotient rule.

2. [1.(x^2) - (x+1)(2x)] / x^4 another basic one, the only thing to note would be that the derivative of (x+1) is 1, just remember that. Simplifying probably won’t be necessary as to actually use the expression above we would have what we need already.

3. [(2x+3).(x+1) - (x^2+3x).1] / (x+1)^2 just remember the differentiated terms of (x^2+3x) is (2x+3) and (x+1) is 1. Again simplifying this would be unnecessary in fact it would just be time consuming considering the impending quadratic in denominator.

4. Due to the quadratic on the top we could expand, separate the terms and simplify to differentiate but that takes longer. So we’re going to use chain rule on the numerator because we don’t have to simplify . ok the answer is. [2.1.(x+1).(2x) - ((x+1)^2)).2] / 4x^2. Nice and simple, chain rule is a life saver. If you’re wondering why there is a 1 multiplied in the very most left term it’s because the differentiation of (x+1) is 1 so I left it in there for knowledge sake.

5. So I threw this one in here as a trick one because we can use quotient rule on the initial expression however simplifying can be done here. (x-1)^2 can be factoised to (x+1)(x-1) which cancels out with an (x+1) on the top to leave you with (x+1)/(x-1) using quotient rule on this leaves you with [1.(x-1) - (x+1).1] / (x-1)^2 which is a lot easier than using the quotient rule on the initial equation.

So thats the quotient rule with some basic questions, I hope you’ve learned something and if you haven’t practise makes perfect, find some more questions and get to it! You’ll get it after a while.

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