## clara1223 one year ago find f''(x) of f(x)=5cos(x^3)

1. clara1223

$f''(x)$ of $f(x)=5\cos(x ^{3})$

2. zepdrix

Second derivative? Oh boy that'll be a bit of a doozy :o

3. zepdrix

Did you find $$\large\rm f'(x)$$ yet?

4. clara1223

I'm not sure what to do with the x^3 inside

5. zepdrix

Chain... rule :o

6. zepdrix

So you expect that it should give you something like this$\large\rm \frac{d}{dx}5\cos(x^3)=-5\sin(x^3)$

7. zepdrix

But chain rule tells us to multiply by the derivative of the inner function.$\large\rm \frac{d}{dx}5\cos(x^3)=-5\sin(x^3)\cdot\color{royalblue}{(x^3)'}$

8. zepdrix

$\large\rm \frac{d}{dx}5\cos(x^3)=-5\sin(x^3)\cdot\color{orangered}{(3x^2)}$Ya?

9. clara1223

ya so we have $-15x ^{2}\sin(x ^{3})$ correct?

10. zepdrix

Looks good. Now things get a little more complicated. Looks like product rule from here, ya?

11. clara1223

ya...

12. zepdrix

Here is how you would "set it up"$\large\rm \frac{d}{dx}-15x^2 \sin(x^3)=\color{royalblue}{(-15x^2)'}\sin(x^3)-15x^3\color{royalblue}{\left(\sin(x^3)\right)'}$And you need to differentiate the blue parts.

13. clara1223

i know that (-15x^2)' is -30x but what is (sin(x^3))'?

14. zepdrix

You should be able to do this one :) We just figured out what (cos(x^3))' is. This one is very very similar.

15. zepdrix

Claraaaaaaaaaaa! Gotta learn that chain rule!! MOAR PRACTICE! :)

16. clara1223

OK so $(\sin(x ^{3}))'=3x ^{2}\cos(x ^{3})$ yes?

17. zepdrix

Good :)

18. zepdrix

$\large\rm \frac{d}{dx}-15x^2 \sin(x^3)=\color{royalblue}{(-15x^2)'}\sin(x^3)-15x^3\color{royalblue}{\left(\sin(x^3)\right)'}$ $\large\rm \frac{d}{dx}-15x^2 \sin(x^3)=\color{orangered}{(-30x)}\sin(x^3)-15x^3\color{orangered}{\left(3x^2\cos(x^3)\right)}$And then maybe combine things a lil bit.

19. clara1223

$-30x \sin (x ^{3})-45x ^{5}\cos(x ^{3})$ yes?

20. zepdrix

Woops! I made a boo boo somewhere!

21. clara1223

where?

22. zepdrix

When I did the setup:$\large\rm \frac{d}{dx}\left[-15x^2 \sin(x^3)\right]=\color{royalblue}{(-15x^2)'}\sin(x^3)\color{red}{-15x^3}\color{royalblue}{\left(\sin(x^3)\right)'}$Do you see anything wrong with this red part?

23. anonymous

zepdrix how do you get those colors for your latex equations?

24. anonymous

dont mean to distract but I am just curious

26. zepdrix

The formatting language is LaTex You can google it and find out a lot of cool little features :) \color{specific color goes here}{your math goes here} for color

27. zepdrix

example: \color{red}{x^2} you would put this manually into the equation tool $$\color{red}{x^2}$$

28. anonymous

Ahh cool thank you... yea I use latex but never with colors but that is handy I will try and use it :D

29. clara1223

@zepdrix I don't see anything wrong with the red part. When I do the product rule I get the same thing... What am I missing?

30. zepdrix

@PlasmaFuzer Another cool thing you can do is: Right click someone's math code > Show Math As > Tex Commands and that will give you a pop up box of how I/someone entered it into the equation tool.

31. zepdrix

It was a -15x^2, ya? Not -15x^3. I musta misclicked the power lol

32. anonymous

Nice!!! thank you again :D :D anyways Ill stop distracting sorry clara cya both later

33. anonymous

still relatively new to site... figuring things out

34. clara1223

Oh! I got -15x^2, didn't even notice

35. zepdrix

So after differentiating, I should've had,$\large\rm \frac{d}{dx}-15x^2 \sin(x^3)=\color{orangered}{(-30x)}\sin(x^3)-15x^2\color{orangered}{\left(3x^2\cos(x^3)\right)}$So we don't end up with a 5th power over there, ya? :)

36. clara1223

$-30x \sin (x ^{3})-45x ^{4}\cos (x ^{3})$ right? because when you multiply powers you add them?

37. zepdrix

sec :) just double checking our work really quick

38. zepdrix

yayyy good job \c:/

39. clara1223

I just multiplied the -15x^2 and the 3x^2 in front of the cos to get -45x^4

40. clara1223

That's the most simplified it's gonna get right?

41. zepdrix

You could pull an x out of each term, but again, unnecessary :) Ya that's a good final answer.

42. clara1223

Thank You!! you have been so helpful. I have 3 more questions tonight, hope you can help :)

43. zepdrix

Calc is so much fun isn't it claire bear? :O