I understand the method used in implicit differentiation, it's just an application of the chain rule. But why can you define y as a function of x?In this equation for example: x 2 + y 2 = 1 2 x + 2 y y ′ = 0Why isn't it just this?: 2 x + 2 y = 0Thank you in advance.

Question

I understand the method used in implicit differentiation, it&#039;s just an application of the chain rule. But why can you define   y as a function of   x?In this equation for example:      x    2    +      y    2    =  1  2  x  +  2  y      y    ′    =  0Why isn&#039;t it just this?:  2  x  +  2  y  =  0Thank you in advance.

Alaina Mcintosh · Accepted Answer

(1).   {  (  x  ,  y  )  :      x    2    +      y    2    =  1  } is NOT the graph of a function.(2). There IS a function   f  (  x  ) for       |    x      |    &amp;lt;  1 such that   f  (  x  ) is differentiable and       x    2    +  f  (  x      )    2    =  1. In fact there are 2 such functions. And we have  0  =      d          d      x        (  1  )  =      d          d      x        (      x    2    +  f  (  x      )    2    )  =  2  x  +      d          d      x        (  f  (  x      )    2    )  =  2  x  +  2  f  (  x  )      f    ′    (  x  )  .Note: In differentiation, it matters what you differentiate BY. Just as in division, where it matters whether you divide by x or by y. The derivative of       y    2   with respect to   y is   2  y  . The derivative of       y    2   with respect to   x is   2  y            d      y              d      x        .

Brandon Monroe · Accepted Answer

I suppose that the use of the implicit function theorem could make things clearer.Consider  F  (  x  ,  y  )  =  0and compute       F    x    ′    (  x  ,  y  ) considering   y as a constant and then F′y(x,y) considering   x as a constant. Now, the implicit function theorem            d      y              d      x        =  −                    F        x        ′            (      x      ,      y      )                      F        y        ′            (      x      ,      y      )

I understand the method used in implicit differentiation, it's just an application of the chain rule. But why can you define y as a function of x? In this equation for example: x^2+y^2=1 2x+2yy′=0 Why isn't it just this?: 2x+2y=0

Open question

Answer & Explanation

New Questions in Calculus 2