Question

Find the derivative of the function.

y = (5x - 4)^4(8x^2 - 5)^-3

Answer 1

Wanna work on this one together?

Answer 2

yes, please :)

Answer 3

Alright! You know about chain rule?

Answer 4

I do, but I still mess it up when trying to solve problems.

Answer 5

Lol, it's cool. I'm only in my second week of cal, so I can't judge.

Answer 6

Alright, you can tell it's going to be a fraction due to the negative power, right?

Answer 7

yes

Answer 8

Alright! What we'll want to do is to use quotient rule.
Can you tell me what the quotient rule is?

Answer 9

(What sucks is that it's difficult to explain while typing out the equations. These types of questions are looongggg and look much better on paper)

Answer 10

finding the derivative of a function that is the quotient of another function

Answer 11

Yup. And do you know what exactly it is? Like the equation itself?

Answer 12

no

Answer 13

Okie dokie.

Answer 14

is it d/dx

Answer 15

It isn't.
But before I explain that, could you rewrite the original equation in fraction form?

Answer 16

Here's your original equation to help:
y = (5x - 4)^4(8x^2 - 5)^-3

Answer 17

(5x-4)^4 / (8x-5)^3

Answer 18

Perfect!

Answer 19

Ok, so here we go:
We will assume the numerator is g(x)
and the denominator is h(x)

Answer 20

then i got this

4(5x-4)^3 (10x-1) / (5-8x)^4

Answer 21

but i'm confused myself how i ended up with (10x-1)

Answer 22

The difference quotient says that the derivative of two functions in the form of:
g(x) /h(x) is equal to this beast: http://upload.wikimedia.org/math/3/8/5/3856f28d4b93877f760d4dbd301176cd.png

Answer 23

Well, we'll see if that's right as we go through this. ^_^
Does that equation I linked to make sense?

Answer 24

somewhat

Answer 25

Alright. If you have pen and paper (hopefully you do, because we're going to use a lot of it), you should write down this stuff:

Answer 26

can you tell me what g(x) is equal to?

Answer 27

Remember:
"Ok, so here we go: We will assume the numerator is g(x) and the denominator is h(x)"

Answer 28

(5x-4)^4= g(x)

Answer 29

perfect! Be sure to write that down.

Answer 30

and what is h(x)?

Answer 31

(8x-5)^3= h(x)

Answer 32

awesome! Now looking back at the link I posted, we're missing two things (not including f'(x), which is the answer we're looking for obviously). What two things are we missing?

Answer 33

Here's the link again:
http://upload.wikimedia.org/math/3/8/5/3856f28d4b93877f760d4dbd301176cd.png

remember, we know g(x) and h(x)

Answer 34

Oh nooooesss... did you run away? :P

Answer 35

no

Answer 36

are we missing the primes

Answer 37

Perfect! And the primes means the derivatives. So here's where the 'fun' begins!

Answer 38

You wanna do g'(x) first or h'(x)?

Answer 39

h'(x)?

Answer 40

okie dokie.

Answer 41

h(x) was defined as (8x^2 - 5)^3

does this sound good so far? (I might have made typoes, so double check please)

Answer 42

yes

Answer 43

Awesome. So let's do this. Wanna take a guess at the first step we must do?

Answer 44

factor? which i'm horrible at :(

Answer 45

Nope, we don't have to do that at all ^_^

Answer 46

So that must be a relief :P

Answer 47

Alright, remember power rule?

Answer 48

oh yea

Answer 49

kinda

Answer 50

Well, quick review:
could you power rule this for me:
x^4

Answer 51

BTW, I need to depart in about 10 minutes. :(

Answer 52

x^4-1= x^3

Answer 53

close. We need to do something more.

Answer 54

oh no... :'(

Answer 55

hint: x^2 will turn into 2x^1

Answer 56

ah..

Answer 57

So what does x^4 turn into?

Answer 58

4x^3

Answer 59

perfect!

Answer 60

Ok I'll remind you of how chain rule works now. :D

Answer 61

ok

Answer 62

Here's what our original problem was:
(8x^2 - 5)^3

Answer 63

With chain rule, you first do powerrule while making believe the stuff inside the parenthesis is your X. However, because the stuff inside the parentheses is NOT X, you have to also derive that stuff next.

I will show an example.

Answer 64

"differentiate (2x)^3"

So... we can agree that (2x) isn't really x

So here's what we do:

1. Make believe it IS x, and do power rule:
3(2x)^2

Answer 65

does this one make sense?

Answer 66

(there's one more step after this, just to warn you. We're not done yet)

Answer 67

kinda

Answer 68

Well, to double check, I'll let you do one:

(5x+4)^3

Could you do the power rule to it like I did?

Answer 69

3(5x)^5

Answer 70

Kinda close.

Answer 71

3(5x)^3

Answer 72

5(3x)^3

Answer 73

(5x+4)^3
a. drop the three up front
3(5x+4)^3

b. lower the 3 exponent by 1 (it's ALWAYS minus 1 no matter what)
3(5x+4)^2

Answer 74

Did that up there make sense?

Answer 75

yes it did

Answer 76

awesome. So I just want to be 100% sure that you've got this:

Do step one of the chain rule to this:

(8x-6)^9

Answer 77

9(8x-6)^8

Answer 78

heck ya! :D

Answer 79

:D

Answer 80

Here's step two of the power rule:
Multiply it by the derivative of the stuff inside the parentheses.
So... let's go ahead and continue the previous example.
9(8x-6)^8

What is the 'stuff inside the parentheses'?

Answer 81

Gotta go real soon, so please answer quickly lol

Answer 82

(72x-54)^8

Answer 83

We can't do that yet unfortunately. Just lemme know what the stuff inside the parentheses are.

Answer 84

(8x-6)

Answer 85

perfect! So step two of the chain rule wants us to multiply step 1, which was 9(8x-6)^8 with the derivative of 'the stuff inside the parentheses'.

So, what is the derivative of the stuff inside the parentheses?

Answer 86

Just the derivative; we aren't going to multiply anything yet.

Answer 87

I don't understand that , sorry

Answer 88

no problemo!

Answer 89

What if we changed it from (8x-6) to just 8x-6
Does that make it less scary?

Answer 90

Like, "derive 8x-6"?

Answer 91

if it still doesn't make sense, just lemme know.

Answer 92

ah.... so confused

Answer 93

It's ok.

Alright, do you remember that any number without a variable turns into 0 when you derive it?

Answer 94

They should have explained that in cal class.

Answer 95

So do you remember that one? You've gotta tell me or I won't know what to explain. :P

Answer 96

I don't remember my professor can't speak english for pellet

Answer 97

Lol. Just write this down:
The derivative of a constant is zero.

Answer 98

ok