Question

f(x)=(5x^6 + 2x^3)^4

Answer 1

I did the chain rule and got: \[f'(x)=4(5x^6+2x^3)^3\times(30x^5+6x^2)\]
But how would we simplify this?

Answer 2

My textbook is giving me an answer that I don't know how it got to that.

Answer 3

What is the answer in your textbook?

Answer 4

It's: \[f'(x)=24x^11(5x^3+2)^3(5x^3+1)\]

Answer 5

24x^11

Answer 6

ok
We have
\[f'(x)=4(5x^6+2x^3)^3(30x^5+6x^2)\]
I want you to notice the first parenthesis
\[(5x^6+2x^3)\]
What can you factor out from this?

Answer 7

We can factor out an x^3

Answer 8

exactly!
\[(5x^6+2x^3)=x^3(5x^2+2)\]
But!! remember, all this is actually CUBED
so we have
\[(5x^6+2x^3)^{3}=(x^3(5x^2+2))^3=??\]

Answer 9

Distribute the cube

Answer 10

don't expand it though, that will make it messy!

Answer 11

Ok so would it be: x^9(5x^3+2)^3? I'm not exactly sure since I usually expand whenever I cube something.

Answer 12

exactly! so we have
\[f'(x)=4x^9(5x^2+2)(30x^5+6x^2)\]
Now factor our as much as u can from the 2nd parenthesis

Answer 13

sorry theres a cube on the 5x^2+2

Answer 14

Oh okay! So the second paranthesis would be: 6x^2(5x^3+1)

Answer 15

And then we just multiply the 4x^9 and 6x^2 together to get 24x^11

Answer 16

yep!\[f'(x)=4x^9(5x^2+2)^3.6x^2(5x^3+1)\]
Multiply them together

Answer 17

yup!!

Answer 18

Cool! And we get the same answer like in the book! Thank you so much! :)

Answer 19

you're welcome!