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MathSofiya Group Title

how is the differentiation of x=ky^2 equal to \[1=2ky\frac{dy}{dx}\]

  • 2 years ago
  • 2 years ago

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  1. Spacelimbus Group Title
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    implicite differentiation, in this case y(x), such that y is a function of x.

    • 2 years ago
  2. Spacelimbus Group Title
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    But that's only one guess in this case, the mechanical way I remember for implicit differentiation is derivate as if you were deriving something in terms of x and then just multiply it by dy/dx, chain rule.

    • 2 years ago
  3. MathSofiya Group Title
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    I"m working on the separable equations section of my differential equations chapter.

    • 2 years ago
  4. Spacelimbus Group Title
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    I will just add this, maybe it helps you, this is the chain rule for multivariable calculus. The above equation you can write like that \[z=f(x,y)=x-ky^2\] So the multivariable chain rule says \[ dz = f_x dx + f_ydy \] \[ \frac{dz}{dx}= f_x+f_y\frac{dy}{dx}\] This is far from a proof, but you can read some application out of it. Implicit differentiation doesn't selectively deal with partial derivatives though.

    • 2 years ago
  5. MathSofiya Group Title
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    The original question reads. Find the orthogonal trajectories of the family of curves x=ky^2, where k is an arbitrary constant. And the first thing they did was differentiate x=ky^2 to get \[1=2ky\frac{dy}{dx}\]

    • 2 years ago
  6. Spacelimbus Group Title
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    the gradient would be orthogonal.

    • 2 years ago
  7. MathSofiya Group Title
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    I haven't learn anything about gradients yet. This is only chapter 9 of stewart's calculus

    • 2 years ago
  8. Spacelimbus Group Title
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    Another relationship for orthogonal functions is \[ m_n \cdot m_y = -1 \] where \(m_n\) is normal to \(m_y\) but I don't see why they apply this sort of differentiation here.

    • 2 years ago
  9. MathSofiya Group Title
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    Oh I think I see what they've done. They rearranged the equation to a separable equation, did the integral. Then stated: THe orthogonal trajectories are the family of ellipses given by the following equation. \[x^2+\frac{y^2}{2}=C\]

    • 2 years ago
  10. MathSofiya Group Title
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    for y=k/x I get ln|y|=ln|-x|+C What do you think?

    • 2 years ago
  11. Spacelimbus Group Title
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    is this the integrated form?

    • 2 years ago
  12. MathSofiya Group Title
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    yep. \[\int \frac1ydy=\int-\frac1x dx\]

    • 2 years ago
  13. Spacelimbus Group Title
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    \[ \large \int \frac{1}{y}dy = - \int \frac{1}{x}dx \] So you can distribute the minus sign and you don't need to carry it inside your logarithmn.

    • 2 years ago
  14. MathSofiya Group Title
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    ok ln|y|=-ln|x|+C

    • 2 years ago
  15. Spacelimbus Group Title
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    perfect.

    • 2 years ago
  16. MathSofiya Group Title
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    Thank you!

    • 2 years ago
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