## ranyai12 2 years ago Can someone please help?! For the function f(x,y)=5(x^6)y+4sin(xy), calculate the indicated partial derivatives f(x)= f(y)= f(xxy)=

1. jtvatsim

You mean $f_x, f_y, \enspace \text{and}\enspace f_{xxy}$ right?

2. jtvatsim

So, for $f_x$ treat y like a constant and take the derivative as usual. In other words, pretend that y is the number 2 or something and do the derivative of x.

3. jtvatsim

So for, $f(x,y)=5(x^6)y+4\sin(xy)$ Treat y like any old number and we get that $f_x=5y\cdot 6x^5 + 4\cdot y \cos(xy)=30x^5y+4y \cos(xy)$ Notice how we used the chain rule for sin(xy) and thought of it just like the derivative of sin(2x) = 2cos(2x).

4. jtvatsim

Does that make sense to you? :)

5. ranyai12

ok so i think i got it but when I attempted for f_y i got 5x^6-4xsin(xy) and it was wrong can you lease tell me what im doing wrong

6. ranyai12

I figured out my mistake but whats fxxy?

7. jtvatsim

OK, so for f_y we will get the following

8. jtvatsim

brb

9. ranyai12

i solved for f_y but I cant seem to figure out what f_xxy is supposed to be

10. jtvatsim

kk

11. jtvatsim

So for f_xxy, first do f_x you will get some equation. Then take the equation you get and do f_x again you will get another equation. Last, take the equation and do f_y and thats the answer.

12. jtvatsim

It's like doing the third derivative, but you take the derivative with respect to different variable each time

13. jtvatsim

$f(x,y)=5(x^ 6 )y+4\sin(xy)$ $f_x = 30yx^5 + 4y \cos(xy)$ $f_{xx} = 150yx^4 - 4y^2 \sin(xy)$ $f_{xxy}=150x^4-4[y^2\cdot y \cos(xy) + \sin(xy)\cdot 2y]$

14. jtvatsim

you must use a product rule for the last part, think of having 4y^2 times sin(2y) you would have to use the product rule.

15. ranyai12

im sorry but can you explain that a little better im still confused

16. ranyai12

@jtvatsim

17. jtvatsim

18. ranyai12

thank you but the answer is wrong :(

19. ranyai12

@agent0smith d you have any idea on how to solve for the last part?

20. zepdrix

Were you able to solve the first two parts? :o

21. ranyai12

yes i did but i cant seem to figure out the last part

22. zepdrix

Yah Jvatsim made a tiny mistake on his last derivative, let's continue from $$f_{xx}$$.

23. ranyai12

ok thank you

24. zepdrix

$\large f_{xx} =\color{royalblue}{150x^4}y - 4y^2 \sin(\color{royalblue}{x}y)$ We're going to be taking the derivative with respect to y for our last step, so we can think of these blue values as constants. The first part shouldn't be too bad, ~A constant times y, taking the derivative with respect to y will leave us with just the constant yes? $\large 150x^4$

25. zepdrix

For the next part, we have two functions of y, so we have to apply the product rule.

26. ranyai12

ok but how would we do that

27. zepdrix

$\large \color{orangered}{(-4y^2)'}\sin(xy)-4y^2\color{orangered}{\left[\sin(xy)\right]'}$ Sorry for the delay, got sidetracked. So here is how we setup the product rule, We need to take the derivative of the orange terms. And we're differentiating these orange parts only with respect to y.

28. zepdrix

$\large (-8y)\sin(xy)-4y^2\color{orangered}{\left[\sin(xy)\right]'}$Understand the first term ok? I guess remembering how to setup product rule is pretty important for this :\

29. ranyai12

yea i do thanks

30. ranyai12

for the second part would you consider x as a constant

31. zepdrix

Yes c: You'll end up applying the chain rule after you take the derivative of the sine portion.

32. zepdrix

So yah the x being a constant is important to remember c:

33. zepdrix

In the previous partials you did, the y was the constant, so an extra y kept popping out each time you took a derivative. The opposite variable will pop out this time

34. ranyai12

so the second part would be xcos(xy)?

35. ranyai12

(−8y)sin(xy)−4xy^2[cos(xy)]

36. zepdrix

yes good.

37. ranyai12

so would that be the answer?

38. zepdrix

Yes, you have 3 parts I believe. The first part, and then the other two parts which came from the product rule.$\large f_{xxy}=150x^4-8ysin(xy)-4xy^2\cos(xy)$ We were working on each part separately, hopefully you didn't lose track of that first part :) heh

39. ranyai12

haha i ddnt forget thank you sooo much!