Taylor Blog #4

This week we learned about the product rule and the quotient rule of derivatives
For both the product rule and the quotient there is a recognizable format that will allow you tho know which rule you will need to use and for each rule there is a formula to memorize and put into effect to find the derivative.
The product rule:
The product rule is recognized as F(x)G(x)
The formula for solving with the product rule is
D/Dx [F(x)G(x)] = F(x) Gprime(x) + G(x) Fprime(x)
Example: (3x-2x^2) (5+4x)
First I take the derivative e of each so that when I plug in I already know what the derivatives are to be plugged in
D/Dx [3x-2x^2]= 3-4x
Therefore F prime= 3-4x
D/Dx [5+4x]= 4
Therefore G prime= 4

Now you plug into the formula
Therefore
3x-2x^2(4)+ 5+4x (3-4x)
Distribute
12x^2x-8x^2+15-20x+12x-16
Simplify
-24x^2 + 4x +15
Because this equation cannot be simplified any further
Dx= -24x^2+4x+15


The quotient rule:
The quotient rule is recognized as F(x)/G(x)
The formula for solving with the quotient rule is
D/Dx [F(x)/G(x)] = G(x) Fprime(x)- F(x) Gprime(x)/ [G(x)]^2
Example: 5x-2/x^2-1
First I take the derivative of each so that when I plug in I already know what the derivatives are to be plugged in
D/dx [ 5x-2] = 5
Therefore F prime= 5
D/dx [x^2 – 1] = 2x
Therefore G prime = 2x
Now you plug into the formula
Therefore
(x^2+1)(5)-[(5x-2)(2x)]/(x^2+1)^2
Distribute
5x^2+5-[10x^2-4x]/(x^2+1)^2
Distribute the negative
5x^2+5-10x^2+4x/(x^2+1)^2
Simplify
-5x^2 + 4x+5/ (x^2+1)^2
Because this equation cannot be simplified any further
DX= -5x^2 + 4x+5/ (x^2+1)^2

Some things to remember:
**don’t forget that when solving with the quotient rule the negative must be distributed to the entire product of f(x)G prime(X)
** don’t forget which order F(X)G prime(X) and G(X)F prime(X) go in for each equation because it DOES matter.