In the next topic in our exploration of derivatives, we will be talking about the concept of the quotient rule. The quotient rule is another method of solving more complex functions.

The quotient rule is used to differentiate a function in which one expression is divided by another. An example of such an equation would be y=((x^3+4x+1)/(x^2-3x)). In some cases, you might be able to divide the function and use the power rule, but in these cases, it might take longer to divide and you might end up with a remainder which would be usually hard to deal with. So, the quotient rule could make the process of finding the derivative easier in these types of functions.

So, What is the quotient rule? Well, first, let's define the top expression as f(x) and the bottom expression as g(x). The formula for the quotient rule is ((f’(x)*g(x))-(g’(x)*f(x)))/(g(x))^2. In this specific formula, you are supposed to take the derivative of f(x) and multiply it by the expression g(x), and then subtract it by the product of the derivative of g(x) and the expression f(x), and then finally divide the entire result by the square of g(x). This might sound like a mouthful, so I will provide you with an example.

For an example of this quotient rule, let's use the equation y=((x^3+4x+1)/(x^2-3x)). Let’s define (x^3+4x+1) as f(x) and (x^2-3x) as g(x). We first multiply g(x) being (x^2-3x) and the derivative of f(x) which is (3x^2+4). Then you subtract the result by the product of f(x) being (x^3+4x+1) and the derivative of g(x) being (2x-3). Then, you divide the difference by the square of g(x) which would be (x^2-3x)^2. The derivative that you should get is (((x^2-3x)(3x^2+4))-((x^3+4x+1)(2x-3)))/((x^2-3x)^2). This should simplify to (x^4-6x^3-4x^2-2x+3)/((x-3)^2(x^2)).

An easy way to remember the formula is through a specific mnemonic saying. This saying is “low dee high minus high dee low over low squared”.Now that we have completed an example together, this saying should make more sense and make it easier for you to calculate using the quotient rule.

One last thing to note is that the quotient rule shouldn’t be used if it is easy to divide. An example of this equation would be y=(x^2+x)/(x). This equation can easily be divided and you can then just simply use the power rule which would be much faster and easier. I hope these examples have helped you to understand the quotient rule and one more method of differentiation as part of your journey through calculus.