Liam Blog Post 4

Hello, I am Professor Carbutt and I will be teaching you about the Product Rule and the Quotient Rule for derivatives today. 

As you will find when you are multiplying and dividing derivatives of functions, one cannot just revert to using the power rule like you do when you are dealing with a single function. If you use the power rule instead of using the product rule or the quotient rule when dealing with multiple functions, your answers will be separated from each other and you will not be able to multiply or divide them together. In order to avoid this, one must use the product rule and the quotient rule when multiplying and dividing derivatives. 

First I will give you both rules as formulas and then we will use them in examples. 

Product Rule: f'(x) * g'(x) = (f'(x)*g(x)) + (f(x)*g'(x))

Quotient Rule: f'(x) / g'(x) = (f'(x)*g(x)) - (f(x)*g'(x))/ (g(x))^2

Example:

First let's try the product rule.

1.) Derive the following function: F(x)= (10x^2)*(2x^5) 

Let's call (10x^2)  f(x) and  (2x^5)  g(x)

We will do this step by step:

 a. First, take the derivative of the first part of the function, (10x^2). Using the power rule, we know the derivative is 20x. 

b. Next, multiply 20x (derivative of f'(x)) by g(x). 20x*2x^5 = 40x^6

c. Next, repeat steps a and b but this time take the derivative of g(x) instead of f(x) and multiply it by the original f(x), which is 10x^2. 

g'(x)= 10x^4, so (10x^4) * ( 10x^2) = 100x^6. 

d. Finally, we take our two answers ( 100x^6 and 40x^6), and we add them together to get a final answer of 140x^6. 

Next, let's try the Quotient Rule.

2.) derive the following function: F(x)= (10x^2)/(2x^5)
Again, Let's call 10x^2 f(x) and 2x^5 g(x)

a. We will repeat the product function in order to do the Quotient Rule, but instead of adding the two products like we did before, this time we will subtract them. Therefore, we will multiply them seperately and get 100x^6 and 40x^6. This time we will do  100x^6-40x^6= 60x^6

b. We are NOT DONE YET! Now, We divide our answer, 60x^6, by the original g(x)^2, Which is (2x^5)^2. 

The answer to the problem would be 60x^6/(2x^5)^2. In this scenario, it is perfectly acceptable to leave your answer in this form, as it is unnecessary to simplify more than this. 

REMEMBER: if you are uncomfortable with the quotient rule, you can always change the original Division problem into a multiplication problem and use the product rule instead! 

EX: 2x/2x^2= 2x*2x^-2

This rule is very useful in everyday calculus, because you are constantly multiplying and dividing derivatives of functions.