Question

??? What in the world? Product rule

Answer 1

Use the product rule to find the derivative
y=(5x^2+3)(4x-5)

Answer 2

My book literally just says random numbers, no explanations.

Answer 3

Product rule =
\[f'(x)g(x) + f(x)g'(x)\]

Answer 4

So you have two functions basically, 5x^2 +3 is one, 4x-5 is the other. So to do the product rule, you basically take the derivative of the first function, 5x^2 + 3 andmultiply it by the second function
(derivative of 5x^2+3)*(4x-5) +
Then add the derivative of the second function multiplied by the original first function. So in total, you would be doing:

(derivative of 5x^2 + 3)*(4x-5) + (derivative of 4x-5)*(5x^2+3)

Answer 5

That seems like a crap ton of work. Is this the best way to do derivs for polynomials that are in parantheses?

Answer 6

More work than the regular way atleast

Answer 7

Pretty much. Multiplying it out and doing power rule term by term would be just as tedius. This isnt a lot of work honestly, given you know how to do regular derivatives.

Answer 8

So I could always just multiply it out?

Answer 9

If you were really that uncomfortable with it. I almost think multiplying it out would give more chances to mess up, though.

Answer 10

Don't multiply it out - the purpose of this question is to help you learn the product rule. Would you want to multiply it out if the function was instead \[\Large y=(5x^2+3)(4x-5)^{12}\] ?

Answer 11

Psymon, how would you find the derivative to 5x^2+3
As in you personally

Answer 12

10x
I mean, it only takes a split-second glance if you know how to do derivatives.

Answer 13

You just have to get used to a couple things. For one, the derivative of a constant is 0. So when you show me 5x^2 + 3, I know that 3 is going away no matter what. Then its just following this rule:

\[\frac{ d }{ dx }x ^{n}=nx ^{n-1} \]So when I see 5x^2, I know im bringing the 2 down and multiplying it andthen lowering the power of 2 by 1. So
\[5x ^{2}\implies (2)5x ^{2-1}= 10x\]

Answer 14

y=(7sqrt(x)+4)x^2

Answer 15

\[y=7(\sqrt{x+4})x^2\]

Answer 16

Another problem or an answer?

Answer 17

A problem :O

Answer 18

Okay so finding the derivative of the inside of the parantheses..

Answer 19

Gotcha. Well, looks like you might as well just rewrite it as:
\[7x ^{2}(\sqrt{x+4})\]

Answer 20

wait

Answer 21

Well, its the same rule, difference is once you follow the rule you end up witha negative exponent.

Answer 22

http://gyazo.com/233d92fec3886b3fdbbe44a1b1e750e8

Answer 23

Sorry, wrote it wrong:O

Answer 24

Ah, okay, slight bit different.

Answer 25

(7sqrt(x+4))x^2

Answer 26

Well, its up to you if you wish to distributethe x^2 in or not, I dont think it makes a massive difference.

Answer 27

I'll not and get better at the other way :)

Answer 28

Alrighty, so back to :
\[f'(x)g(x) + f(x)g'(x)\]
So first is the derivativeof (7sqrt(x) + 4) multiplied by x^2 Now as I mentioned before, any random floating constant disappears when you take the derivative. So that 4 will just disappear. The 7 is being multiplied by the variable, so that doesnt just disappear like the x does. So either way, we just follow the power rule written above:

\[7\sqrt{x}+4\implies7x ^{\frac{ 1 }{ 2 }}+4\]So the 4 will go away, the 1/2 will come down and then the power will be lowered by one.

\[(\frac{ 1 }{ 2 }*7(x)^{-\frac{ 1 }{ 2 }})x ^{2} \]. So now I have to do (derivative of x^2)*(7sqrt(x) + 4)
Derivative of x^2 is 2x, which is then multiplied by the square root expression we have. So in total, we get:
\[(\frac{ 1 }{ 2 }*7x ^{-\frac{ 1 }{ 2 }})x^{2} + 2x(7\sqrt{x}+4) \]

Now its just how far do you need to simplify it. Depending on the programor the professor, this may be a fine answer or it may have to be simplified as much as possible.

Answer 29

I'll need it simplified as much as possible:(

Answer 30

Never any fun. Alright, so do you know how to handle the negative exponents?

Answer 31

Well normally you just put it as 1/sqrt(equation)

Answer 32

Right. Just making sure. So let's rewrite what we can then:

\[\frac{ 7x ^{2} }{ 2\sqrt{x} }+2x(7\sqrt{x}+4)\] So Imgoing to make this into one fraction and see if it lets me cancel some things out.
\[\frac{ 7x^{2}+4x ^{\frac{ 3 }{ 2 }}(7\sqrt{x}+4) }{ 2\sqrt{x} } \]
\[\frac{ 7x ^{2}+28x ^{2}+16x^{\frac{ 3 }{ 2 }} }{ 2\sqrt{x} } \]

\[\frac{ 35x^{2} +16x ^{\frac{ 3 }{ 2 }} }{ 2\sqrt{x} }\implies \frac{ 35x ^{\frac{ 3 }{ 2 }}+16x }{ 2 } \]

Dunno if you can dumb it down more than that xD

Answer 33

Okay so for this
y=4x^4+8 over x^2

Answer 34

So the bottom, is easily 2x.

Answer 35

4x^4, 4*4x^3

Answer 36

16x^3/2x?

Answer 37

This requires a new rule, quotient rule. Now, this is a problem that you can kind of cheat and do product rule with, too. But striahgt out quotient rule is:
\[\frac{ f'(x)g(x) - f(x)g'(x) }{ [g(x)]^{2} }\]

Answer 38

So yeah, you cant just do the derivative of the top and then the derivative of the bottom, it has its own rule.

Answer 39

Well, that just looks confusing. Why can't you do what I just did? I know it's wrong.

Answer 40

Okay so please walk me through this :P

Answer 41

Yeah, it can be confusing. But it still is just plugging in the numbers and such. So we have 4x^4 + 8 and we have x^2. So the first part of the formula is f'(x)g(x). SO we can take the derivative of the first part pretty easily. The 8 disappears and 4x^4 follows the regular power rule, making it 16x^3, which gets multiplied by x^2 to give us 16x^5 for the firstpart. Now we have a minus sign in between and we have (4x^4+8) multiplied by the derivative of x^2, so just 2x. The final component is just the (g(x))^2 on bottom. So x^2^2 is just x^4. So with all of that, ill put all the values into their appropriate spots in the formula:
\[\frac{ 16x^{5}-2x(4x^{4}+8) }{ x^{4} }\implies \frac{ 16x^{5} - 8x^{5}-16x }{ x^{4} }\implies \frac{ 8x ^{4}-16 }{ x^{3} }\]
Probably not much to do beyond that unless you wanted to factor an 8 out of the top

Answer 42

I'm going to do the rest myself, then I'll be back to do my quizzes with your help?:)

Answer 43

Yeah, sure. Not sure how much longer im awake, but Ill do what i can :3

Answer 44

http://gyazo.com/469457671a6dd0b36013e4588a992220

Answer 45

Well, gotta be able to do the derivative first no matter what :P

Answer 46

I got the answer but it's saying it's wrong..

Answer 47

whatd you get for the derivative?

Answer 48

I got..
y=-4x^2+4/(x^2+1)^2