Quotient Rule: A Shortcut That Doesn't Require to Memorize Another Formula

The quotient rule is a formula that lets you calculate the derivative of quotients between functions. It is a more complicated formula than the product rule, and most calculus textbooks and teachers would ask you to memorize it.

I don't think that's neccesary. So, I'll show you an alternative way of solving problems involving quotients that doesn't need the memorization of yet another formula. 

The method I use is a combination of chain rule+product rule, which you already know. And I'll show you the method directly with examples. Let's begin.

Example 1

Let's say we want to find the derivative of:

Here we have the quotient between two functions. To find the derivative of this function, we only need to remember that a quotient is in reality a product. So, the first thing we do is to write the function as a product, which we can do like this:

Now that we have a product, we can apply the product rule. First we determine the functions u and v:

And we invoke the product rule formula:

And with some algebra we get the following expression:

And that's it. I think that it is more practical to learn to solve quotient problems like this than memorizing the quotient rule. Even if you can memorize the quotient rule now, you won't remember it 1 year from now, I assure you.

Example 2

Let's find the derivative of:

Again, we can write this as:

And here we apply the product rule. We determine our u and v:

And apply the formula:

And that equals:

And you can play algebraically with it, but that's basically the answer.

Example 3

Let's figure out the derivative of:

We'll use the same technique:

And we apply our trusted product rule:

And the answer is:

That's enough quotient rule in disguise for now. Your next step should be to learn about the derivative of ln(x).

Conclusion

  • There's no need to memorize yet another formula.
  • You can consider the quotient rule as a special case of the product rule.
  • Whenever you have a function with a quotient, you can always write it like a product.

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