Question

The Product and Quotient Rule:

Calculate
dy
dx

You dont need to expand answer

y = (5x^2+x)(x-x^2)

Answer 1

how much can you do?

Answer 2

not sure how to get the derivative
-- im guessing the next step goes like this.

11(5x^2\[11(5x ^{2}+x)+(x-x ^{2})\]

Answer 3

the easiest way to do this is to expand the terms and get a polynomial

y = -5x^4 + 4x^3 - x^2

Answer 4

then what would be the next step to that?

Answer 5

Product rule: \[\frac{\text{d}}{\text{d}x}f(x)g(x) = f'(x)g(x) + f(x)g'(x)\]So derivative first factor and left second one untouch, then derivative second one and left first one untouch.

Answer 6

now you can differentiate without the use of the product rule

Answer 7

@campbell_st Would be better if s/he learn the product rule. Just saying.

Answer 8

ok then how would i find the derivatives?

Answer 9

what is f'(x), g'(x)?

Answer 10

Derivative of the function. f(x) = (5x^2+x) and g(x) = (x-x^2)

f'(x) = d/dx (5x^2+x) and g'(x) = d/dx (x-x^2)

Does this make sense?

Answer 11

well the product rule can lead to mistakes... I think expanding makes it easier as most people are more confident with straight differentiation.

Answer 12

so it would be (5x^2+x)(10)+(x-x^2)(2)?

Answer 13

@campbell_st You have a point, but it actually depends on problem. If you face something like this: \[\Large \left(2x^3 + \frac{1}{\sqrt{x}}\right)^4 (3x^2 + x^{\frac{1}{4}})\] Do you think expanding this is easier than using product rule (and power rule)?

Answer 14

No. First, derivative first factor (5x²+x), and left second one (x-x²) untouched.
Do you know how to derivative (5x²+x)?

Answer 15

\[\frac{dy}{dx} = -20x^3 + 12x^2 - 2x\]

is the derivative...

so just answer the questions on there merits... for me... expand then differentiate will work.

it can be written \[\frac{dy}{dx} = -2x(10x^2 - 6x + 1)\]

Answer 16

@chrislb22 Do you want to use product rule or just do the campbell_st's way? This method does make it easier, but I think it'd be better for you to learn the product rule, just in case?

It's up to you, though.

Answer 17

product rule

Answer 18

most likely i will be tested using the product rule, but in campbell's case it does make a bit more sense

Answer 19

Well, just like I said, factor (5x²+x) and left (x-x²) untouched. Can you derivative (5x²+x)?

Answer 20

derivative of (5x^2+x) is 11?

Answer 21

10 from 5x^2 then 1 from x?

Answer 22

Actually, while derivative the function, you basically subtract the power by one. So the derivative of 5x² is 10x. And, obviously, the derivative of x is one. So the derivative of 5x+x is 10x + 1.

Answer 23

Does this make sense?

Answer 24

oh i got you, i understand that now

Answer 25

so the derivative for x-x^2 is 1-2x?

Answer 26

Yeah, derivative first one and left second one untouched then "put in" plus sign then derivative second one and left first one untouched. Does this help?

Answer 27

"so the derivative for x-x^2 is 1-2x?"

Yes!

Answer 28

Sorry, I accidentally deleted your message, what is your solution again?

Answer 29

he Product and Quotient Rule:

Calculate
dy
dx

You dont need to expand answer

y = (5x^2+x)(x-x^2)

Answer 30

No, no, I mean your answer.