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Implicit Differentiation

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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

I don't know how to go from here to find the m. Please help :)

  • Implicit Differentiation -

    Your derivative equation looks good.

    sub in x=1 and y = -5/31 into that derivative equation, then solve it for y'
    that will be your m
    Good luck with that messy arithmetic.

  • Implicit Differentiation -

    I got y'=(2x+x^2+12yx+6y^2-y)/(x+6y+6)
    But when i plugged in the numbers it came out right... do you think the equation is wrong or i just typed in the numbers wrong?

    BTW thanks for replying!

  • Implicit Differentiation -

    using your [(x+6y)(y')-(y)(1+6y')]/(x+6y)^2=2x
    we can continue
    xy' + 6yy' - y - 6yy' = 2x
    xy' - y = 2x
    y' = (2x+y)/x
    using the given point:
    y' = (2 - 5/31)/1 = 57/31

    y + 5/31 = (57/31)(x-1)
    31y + 5 = 57x - 57
    57x - 31y = 62

    check my arithmetic.

  • Implicit Differentiation -

    I did this exactly with this question and i got the wrong answer.

    Use implicit differentiation to find the slope of the tangent line to the curve

    y/(x+5y)=x^3+5
    at the point (x=1, y=6/–29).

    But I noticed after "we can continue..."
    the denominator was moved to the right side of the equation.. so maybe that's it..??

  • Implicit Differentiation -

    Sorry. I meant the denominator from the quotient rule was never multiplied to the right or so. I don't see it. Where did that portion go?

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