## mathcalculus Group Title help please: Use implicit differentiation to find the equation of the tangent line to the curve at the point. (attached below) 11 months ago 11 months ago

1. mathcalculus Group Title

2. mathcalculus Group Title

i know i have to use implicit differentiation, then plug into derivative to find slope, plug slope and given points to point-slope formula.

3. mathcalculus Group Title

Problem I have is step 1

4. phi Group Title

take the derivative with respect to x. use the product rule for example, on the xy term you do this $\frac{d}{dx}xy = x \frac{d}{dx}y+ y \frac{d}{dx}x \\ = x \frac{dy}{dx}+ y \frac{dx}{dx} \\ = x \frac{dy}{dx}+ y$ use the same idea on $$xy^3$$

5. mathcalculus Group Title

i got x(3y*dy/dx) +y^3

6. mathcalculus Group Title

@phi

7. mathcalculus Group Title

so.. 3xy*dydx +y^3

8. mathcalculus Group Title

dy/dx**

9. phi Group Title

you mean 3 x y^2 dy/dx for the first part d/dx of y^3 is 3 y^2 dy/dx

10. mathcalculus Group Title

okay im lost with the second part.

11. mathcalculus Group Title

i know that you took the derivative of xy^3= 3xy^2

12. mathcalculus Group Title

+xy= 1?

13. mathcalculus Group Title

*dy/dx

14. phi Group Title

from the beginning $\frac{d}{dx}\left(xy^3 + xy = 4\right) \\ x \frac{d}{dx}y^3 + y^3 \frac{d}{dx}x+ x\frac{d}{dx}y+ y \frac{d}{dx}x=\frac{d}{dx}4$ that simplifies to $x \frac{d}{dx}y^3 + y^3+ x\frac{dy}{dx}+ y =0$ now you can do the d/dx of y^3 , right ?

15. mathcalculus Group Title

yes.

16. phi Group Title

use the power rule $\frac{d y^n}{dx} = n y^{n-1} \frac{dy}{dx}$

17. mathcalculus Group Title

this isn't applying chain rule or is it? because i understand what youre doing here makes sense/

18. phi Group Title

yes, it is the chain rule

19. phi Group Title

normally you do d/dx of x^3 and you get 3x^2 dx/dx but the dx/dx is left off and the rule looks like $\frac{d\ x^3}{dx} = 3 x^2$

20. mathcalculus Group Title

okay, let me give this a try.

21. mathcalculus Group Title

okay I got this down: x*3y^2(dy/dx) +3y +x(dy/dx)+y=0

22. phi Group Title

you mean y^3 not 3y , right ?

23. mathcalculus Group Title

oh so only find the derivative of [y^3]...

24. mathcalculus Group Title

thanks btw! :)

25. mathcalculus Group Title

then i should just plot the points for x and y

26. mathcalculus Group Title

wait a min. what should i do with dy/dx?

27. phi Group Title

solve for dy/dx and then sub in for x and y to get the value of dy/dx at the point (2,1)

28. mathcalculus Group Title

how?

29. phi Group Title

use algebra

30. mathcalculus Group Title

question: is it possible to divide this: -4x/-2y ????

31. mathcalculus Group Title

how the hell can someone divide this and get 2x/y?????

32. mathcalculus Group Title

33. mathcalculus Group Title

@asnaseer

34. phi Group Title

what part is confusing ? you have $3 x y^2\frac{dy}{dx} + y^3+ x\frac{dy}{dx}+ y =0$ plug in x=2 and y =1 and solve for dy/dx (which is the slope)

35. phi Group Title

at the point (2,1)

36. mathcalculus Group Title

no no, i know that. i was just looking at an example and i see that they divide -4x/-2y and got 2x/y

37. mathcalculus Group Title

it frustrated me to see this and not know how they even possibly do it.

38. phi Group Title

are you asking how to get from $\frac{-4x}{-2y} = \frac{2x}{y}$ ? -4/-2 is +2

39. mathcalculus Group Title

yes but with diferet variables?? 0.o

40. mathcalculus Group Title

different*

41. phi Group Title

what is the question ?

42. mathcalculus Group Title

oh the slope is -1/4! =]

43. phi Group Title

yes. also, the tangent line goes through point (2,1) so you can find the intercept.

44. mathcalculus Group Title

cool, how do you know this?

45. phi Group Title
46. mathcalculus Group Title

okie doke

47. mathcalculus Group Title

thanks @phi you helped me so much! ^_^ so thankful

48. phi Group Title

they asked you to find the tangent line through point (2,1) you found its slope = -1/4 you know it goes through 2,1 so in point-slope form you have y- 1 = (-1/4)(x-2) which you can change to slope-intercept form

49. mathcalculus Group Title

y=-1/4x+3/2

50. phi Group Title

yes