## anonymous one year ago If the derivative of ​f[x_] = (x^2 - 2 Log[x]) ​f'[x_] = (2x - 2/x) Then why is the derivative of ​f[x_] = (x^2 - 2 Log[x])/2 ​f'[x_] = (2x - 2/x)/2 What is the principle that allows the /2 which appears to be some kind of constant to remain in the derivative ?

1. dan815

because derative is a linear operator

2. hartnn

constants can be factored out of the derivative

3. UsukiDoll

wow this guy must be rich to do a QH question

4. dan815

​f[x_] = 2*(x^2 - 2 Log[x]) heres another example this is same as ​f[x_] = (x^2 - 2 Log[x]) + (x^2 - 2 Log[x]) there are 2 of them, and if u remember we can differentiate each term separately

5. anonymous

no just desperate, lol .. I've given up caring about money

6. anonymous

so I just didnt simplify it enough?

7. dan815

no its not like that its just that the derivative is a linear operator

8. UsukiDoll

@plasmataco this is calculus ii material

9. dan815

you can see taking a derivative is a linear operator if you write out the first principles

10. UsukiDoll

A moderator is present in this conversation. Do not go off topic and break the rules

11. dan815

|dw:1434803107603:dw|

12. UsukiDoll

maybe we can try to take the derivatives together.. one is straight forward first it's just derivative and then second is a simple product rule .

13. dan815

|dw:1434803242178:dw|

14. hartnn

/2 is division by 2. multiplication or division of a constant to the function, multiplies or divides its derivative by the same constant.. $$\Large if ~ ~f'(x) = g(x), \\ then , ~~ [a \times f(x)]' =a \times g(x)$$

15. anonymous

ah ok.. that's the piece I was missing hartnn

16. anonymous

even though its a /2 .. it's a constant multiple.

17. hartnn

exactly!

18. anonymous

ah ok.. that makes things click for me.

19. anonymous

sorry dan.. that was a colossal effort, much appreciated.. you lost me a bit there though :)

20. dan815

thats all there is to it, i was showing you why, you can factor the constant out

21. UsukiDoll

that's why he's the honorary professor of Mathematics XD

22. dan815

lol :>

23. anonymous

I got to give this one to hartnn though.. that was the piece I was missing.. I knew the /2 had to carry into the derivative, but I could not work out for the life of me, what rule was making that happen.. once I realized it was just a 0.5 f[x] constant multiple, then it clicked..

24. Plasmataco

I gave mine to dan, so... yeah, give to @hartnn

25. UsukiDoll

mine is also given to dan

26. hartnn

thanks hugh! Its been years i didn't care about medals and stuff, so feel free to give it to anyone you like :)