## mathcalculus Group Title can someone explain to me Implicit Differentiation: if y^3 = 25x^2, determine dx/dt when x = 5 and dy/dt = 1 11 months ago 11 months ago

1. mathcalculus Group Title

these are the steps but i don't understand them :/ 3y^2*dy/dt = 50x*dx/dt When x = 5, y^3 = 25*5^2 y^3 = 625 y = cube root 625 = 8.55 3(8.55)^3*1 = 50*5*dx/dt 1875 = 250dx/dt 7.5 = dx/dt

2. mathcalculus Group Title

isn't y=2.92402?

3. mathcalculus Group Title

i did this on the calculator: sqrt (625)^(1/2)

4. mathcalculus Group Title

(1/3)**

5. mathcalculus Group Title

@satellite73

6. agent0smith Group Title

Let's start with this part... $\Large \frac{ d }{ dt } y^3$use the chain rule to differentiate it... so bring down the exponent, reduce the exponent by one, then multiply by the derivative of y w. respect to t $\Large 3 y^2 *\frac{ dy }{ dt }$

7. mathcalculus Group Title

so I can't replace the dy/dt by 1 since it's given?

8. mathcalculus Group Title

@agent0smith

9. agent0smith Group Title

Well yes but i was seeing if you understood the differentiation... don't worry about plugging in numbers till later.

10. mathcalculus Group Title

and 3y^2 is already the derivative, why do we need to use the chain rule?

11. mathcalculus Group Title

@agent0smith

12. agent0smith Group Title

Because you have to multiply by the derivative of y with respect to t.

13. agent0smith Group Title

3y^2 is NOT the derivative of y^3, UNLESS you're just differentiating with respect to y.

14. tpmys Group Title

normally we have y dependent on x here y, and x both are dependent on some t, so we dont know what the function is so instead of y^3 >>> 3y^2 we have the chain rule y^3 >>> 3y^2 * y'

15. agent0smith Group Title

It's the same with dy/dx... the derivative of y with respect to x is not just 1... it's 1*dy/dx

16. mathcalculus Group Title

ok... :/ it's a little confusing..

17. ranga Group Title

agent0smith explained it nicely: $\frac{ d }{ dy }(y ^{3}) =3y ^{2}$ But $\frac{ d }{ dt }y ^{3} = \frac{ d }{ dy }y ^{3}\frac{ dy }{ dt } = 3y ^{2}\frac{ dy }{ dt }$

18. agent0smith Group Title

It's the same thing with the chain rule $\huge \frac{ d }{ dx } f(x)^n = n*f(x)^{n-1} * f \prime (x)$ you have to always multiply by the derivative at the end.

19. mathcalculus Group Title

why can't we plug in the numbers right away if it's given?

20. agent0smith Group Title

You can... you're probably just better off not doing it until you really know what you're doing.

21. mathcalculus Group Title

right i understand that... @ranga

22. mathcalculus Group Title

well i was taught to plug in.. that's why.

23. mathcalculus Group Title

that's the confusing part.

24. agent0smith Group Title

Fair enough... but you can't plug in until you have this step: 3y^2*dy/dt = 50x*dx/dt

25. mathcalculus Group Title

exactly what i have right now.

26. mathcalculus Group Title

so dy/dt is 1...

27. mathcalculus Group Title

in the steps above, why is this: 3(8.55)^3*1 = 50*5*dx/dt?

28. mathcalculus Group Title

y=8.55? y is suppose to be 2.92402.

29. mathcalculus Group Title

and it's raised to the 3rd, and why not 2?

30. agent0smith Group Title

When x = 5, y^3 = 25*5^2 y^3 = 625 y = cube root 625 = 8.55 i think the next line after that is a mistake and it should by to the power of 2

31. mathcalculus Group Title

so y=2.92402 is correct?

32. agent0smith Group Title

y^3 = 625 y = cube root 625 = 8.55

33. mathcalculus Group Title

how?

34. mathcalculus Group Title

i did this in the calculator: sort(625)^(1/3)

35. mathcalculus Group Title

sqrt*

36. agent0smith Group Title

And you'll get 8.55... 2.94 cubed is close to 3^3 which is 27. Not 625.

37. mathcalculus Group Title

i swear im getting 2.92402

38. mathcalculus Group Title

i plugged it in exactly like iwrote it.

39. agent0smith Group Title

sort(625)^(1/3) why are you taking the square root of a cubed root...

40. agent0smith Group Title

y^3 = 625 y = 625^(1/3)

41. mathcalculus Group Title

OHHH

42. mathcalculus Group Title

i thought the sqrt could do cube root.

43. mathcalculus Group Title

if we did the exponent also.

44. agent0smith Group Title

That would mean ((625^(1/3))^(1/2) = 625^(1/6)

45. mathcalculus Group Title

thank you! i got it !