Implicit Differentiation

posted by .

Find the slope of the tangent line to the curve (a lemniscate)

2(x^2+y^2)^2=25(x^2–y^2) at the point
(–3,1) .

I know that we are supposed to use chain rule for this problem but i think i am messing up on the algebra. Can someone please help me, i keep on getting m= (42x^3-42y^2x)/(58x^2+58y^2)

  • Implicit Differentiation -

    2(x^2+y^2)^2=25(x^2–y^2)
    2(x^2+y^2)^2-25(x^2–y^2)=0
    Expand
    2x^4+4x^2y^2+2y^4-25x^2+25y^2=0
    Differentiate (implicitly) with respect to x, denote dy/dx as y'
    8x^3+8xy^2+4x^2(2y)y'+8y^3y'-50x+50yy'=0
    Collect terms and solve for y' in terms of x and y:
    y'=(50x-8x^3-8xy^2)/(8x^2y+8y^3+50y)
    At (-3,1), x=-3, y=1
    y'=(-150+216+24)/(72+8+50)=9/13

    Check for arithmetic errors.

  • Implicit Differentiation -

    I get
    4(x^2 + y^2)(2x + 2yy') = 50x - 50y y'
    .....
    ....
    y' = ((50x - 8x^3 - 8xy^2)/[ (8x^2)y + 8y^3 - 50y)

    plug in your point (-3,1) to get the slope

    Take it from there.

  • Implicit Differentiation -

    Oh my goodness, thank you soooo much for writing it out i totally wasn't getting it! THank you so much :)

    Do you think you coul check my arithmatic for another problem? It's
    Use implicit differentiation to find the slope of the tangent line to the curve

    y/x+6y=x^2–6 at the point (1–5/31)

    I got [(x+6y)(y')-(y)(1+6y')]/(x+6y)^2=2x then y'=(2x+x^2+12yx+6y^2-y)/(x+6y+6). Is that right?

  • Implicit Differentiation -

    y/(x+6y) or (y/x)+6y?
    As it is, the expression means the latter.

    Since I see (x+6y)y' somewhere, I think you mean the first interpretation.


    y/(x+6y)=x^2–6
    Cross multiply to get
    y=(x^2-6)(x+6y)
    y'=2x(x+6y)+(x^2-6)(1+6y')
    I get
    y'=-(12xy+3x^2-6)/(6x^2-37)
    Check my arithmetic and differentiation and take it from here.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. calculus

    Use implicit differentiation to find the slope of the tangent line to the curve (y/x+8y)=x^8 +9 at the point (1,10/-79).
  2. calculus

    Alright so implicit differentiation is just not working out for me. Use implicit differentiation to find the slope of the tangent line to the curve at point (4,1). y / (x-2y) = x^3 + 4 Tried quotient rule to get the derivative of the …
  3. CAL

    use implicit differentiation to find the slope of the tangent line to the curve (y)/(x-7y)=x^6+5 at the point (1,6/43)
  4. math

    use implicit differentiation to find the slope of the tangent line to the curve (y)/(x-7y)=x^6+5 at the point (1,6/43)
  5. Calculus

    (y/(x+9y))= (x^2)-7 at the point (1, -6/55) Use implicit differentiation to find the slope of the tangent line to the curve
  6. Implicit Differentiation

    Use implicit differentiation to find the slope of the tangent line to the curve y/x+6y=x^2–6 at the point (1,–5/31) . Again i think i'm messing up with the algebra here. I used quotient rule to get [(x+6y)(y')-(y)(1+6y')]/(x+6y)^2=2x …
  7. Calculus ap

    Use implicit differentiation to find the slope of the tangent line to the curve XY^3 + XY = 5 at the point (3,2) .
  8. Math

    Use implicit differentiation to find the slope of the tangent line to the curve y/( x + 4 y) = x^4 – 4 at the point (1,-3/13)
  9. Math

    Use implicit differentiation to find the slope of the tangent line to the curve y/( x + 4 y) = x^4 – 4 at the point (1,-3/13)
  10. Math (Implicit Differention)

    use implicit differentiation to find the slope of the tangent line to the curve of x^2/3+y^2/3=4 at the point (-1,3sqrt3)

More Similar Questions