Comparing the FJ rule with the quotient rule

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Hi, Good morning,

Shall we start right away?

To compare, let’s apply the quotient rule to our example first.

If you may recall, using the FJ rule can reach the answer with one fewer step. Time is critical in exam. So don’t underestimate how much having just one fewer step can help you.

Next, let’s have a closer look to understand the other benefit:

The one step saving comes from the factorization because the FJ rule does it automatically for you.

Besides that, you now no longer need to care who minus who. The minus sign comes from the power of the bottom term and this has been factored in the FJ rule formula. So with FJ rule, you don’t need to worry for the mistake of putting the terms in the wrong order. Hence, lower chance of careless mistake.

This is not all. In my next post on H2 math, I will show how the FJ rule is a better rule than the product rule in product differentiation. All because of the built-in factorisation of the FJ rule.

So, stay tuned!

Introducing the FJ rule

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Hi,

In H2 math, differentiation is one of the cornerstones that every student must master. After discarding the quotient rule in my previous post, let’s fill the void with an enhanced product rule. For now, just call it the FJ rule first.

I will first introduce how the rule looks, then how it works, followed by an example and a comparison with the conventional product rule and quotient rule.

1. Introducing the FJ rule:As I underlined, it looks exactly the product rule we have known since O Levels A Maths, except for the powers. This makes it easier to remember.It works like this:
   • Copy exactly the two original terms down, but with their powers each minus 1;
   • Then the power m times the derivative of the first term f(x) times the second term g(x);
   • Plus the power n times the derivative of the second term g(x) times the first term f(x);
   • These two add together to times the original terms (with power down by 1).

 

2. One working example

You can further factorise this, but it’s enough to show how to apply the FJ rule. Comment if you find it difficult to understand.

In this example, I took 3 steps to reach the answer. In my next post on H2 math, I will apply the conventional product rule and quotient rule to this same example to show that this FJ rule indeed reduces the number of steps and the chances of careless mistakes.

You may try it yourself with different rules now. If not, stay tuned for the next post.

 

best,

FJ

Forget the quotient rule!

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As the first blog entry on H2 maths, let’s start with something simple: quotient rule and product rule. After my sharing, you will see the reason why you should discard the quotient rule.

This is the standard quotient rule:

 

• The bottom times the derivative of the top
• minus
• The top times the derivative of the bottom
• All over
• The bottom squared

The problem students have is the “minus” and to a smaller extent, the “squared“. And the formula is ugly!

Now let’s have a look at our prettier friend, the product rule:

If you notice, the numerator of the quotient rule looks similar to the product rule, except for the minus sign.

So this is what I propose: a hybrid rule combining the two rules together:

With this, you can do away with the ugly quotient rule formula and treat every case using the product rule. This avoids the frequent mistake of forgetting the minus sign in the quotient rule. In addition, you need not memorise which derivative comes first to minus the other derivative.

More importantly, in my next post on H2 math, I will show how treating quotient as products can substantially save time and at the same time, reduce the chances of careless mistakes.

best,