The quotient rule helps you to find the derivative of a function that consists of two functions divided by each other. We can prove it from first principles, but there’s another way to arrive at the rule by implicite differentiation. It goes like this.

Let’s look at the function

Now we take the natural logarithm (but it could be any logarithm) of both sides.

As per the rules of logarithms, we know that

so we can write

Now let’s find the derivative of using implicite differentiation and the chain rule:

or

Now let’s create equal denominators:

Multiply both sides by and we get

Remember that we know

so

Eliminating in the numerator and denominator brings us at

and we’re done:

It’s not much, but for those in need it might be just enough. 🙂