## amy0799 one year ago The function f and g are differentiable at x=10 and x=20 and f(g(x))=x^2. if f(10)=5, f'(10)=4, f'(20)=-5, and g(10)=20, what is the value of g'(10)?

1. freckles

have you used chain rule to differentiate f(g(x)) yet?

2. amy0799

I learned how to do the chain rule

3. freckles

ok then what is $\frac{d}{dx}f(g(x))=?$

4. amy0799

f(x)g'(x)+g(x)f'(x)

5. freckles

that is the product rule we don't have f*g we have f composed with g

6. freckles

this is why I asked you to use chain rule to differentiate f(g(x))

7. amy0799

I thought u use the chain rule when its something to a power

8. freckles

power rule is what you use when you have a constant power

9. freckles

chain rule is what you use when you have a function inside a function

10. amy0799

ooh ok. So how would I do the chain rule for this? would it be f'(g(x))*g'(x)?

11. freckles

that is right

12. freckles

$f(g(x))=x^2 \\ \text{ differentiating both sides } \\ f'(g(x)) \cdot g'(x)=2x$

13. freckles

not enter in 10 for x

14. freckles

$f'(g(10)) \cdot g'(10)=2(10)$

15. freckles

you are given g(10)

16. freckles

g(10)=20 right?

17. freckles

so replace g(10) with 20 $f'(g(10)) \cdot g'(10)=2(10)$ $f'(20) \cdot g'(10)=2(10)$

18. freckles

see if you can finish the rest

19. amy0799

g'(10)=25?

20. freckles

$f'(20)=-5 \\ \text{ so we have } \\ -5 \cdot g'(10)=2(10)$

21. freckles

hmmm how did you get g'(10)=25?

22. amy0799

23. freckles

you do know the operation between -5 and g'(10) is multiplication and not addition :p

24. amy0799

g'(10)=-4?

25. freckles

20/-5 is -4 good worrk

26. amy0799

thank you!

27. amy0799

I have another question, do u mind helping me till?

28. freckles

I can take a look

29. amy0799

In the table below, the values of f(x), g(x), f'(x) and g'(x) are given 2 values of x. if y =[f(2x)+g(x)]^2, find y'(3)

30. freckles

well we know we are going to have differentiate since we want to find y'(of something)

31. amy0799

|dw:1441749521965:dw|

32. freckles

again here you will have to use chain rule this chain rule does come with power rule because we have have a constant power we will also have to use sum rule then chain rule again

33. amy0799

2(f(2x)+g(x))*(f'(2x)+g'(x))

34. freckles

ok I only have one complaint...

35. freckles

let's look at f(2x)

36. freckles

notice the inside function is 2x

37. freckles

the outside function is f( )

38. freckles

when you find the derivative of f(2x) w.r.t. x you should get 2*f'(2x) derivative of inside time derivative of outside

39. freckles

$y(x)=[f(2x)+g(x)]^2 \\ y'(x)=2[f(2x)+g(x)] \cdot [2 f'(2x)+g'(x)]$

40. freckles

now we want to find y'(3)

41. freckles

so replace x with 3

42. freckles

$\\ y'(3)=2[f(2\cdot 3)+g(3)] \cdot [2 f'(2 \cdot 3)+g'(3)]$

43. amy0799

y'(3)=112

44. freckles

$f(6)=? \\ g(3)=? \\ f'(6)=? \\ g'(3)=?$ you found these values from the table ?

45. freckles

$f(6)=-3 \\ g(3)=-1 \\ f'(6)=-2 \\ g'(3)=-5$ this is what I see for those values is that what you also see

46. amy0799

oh I know what I did wrong. Hold on

47. amy0799

y'(3)=72?

48. freckles

purrrfect (you know like a cat)

49. amy0799

haha thank you so much for the help!

50. freckles

np I hope everything makes more sense

51. amy0799

Yup it does thanks to your help :D

52. freckles

np I must go now good luck with calculus

53. amy0799

ok. thanks, I need all the luck I can get haha