For finding the derivative, d y d x of an implicit function, there is this formula,When f ( x , y ) = 0, d y d x = − Differentiation of f w.r.t to x keeping y constant Differentiation of f w.r.t to y keeping x constant I want to understand how this method came about.Is there a clear proof for this formula?

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For finding the derivative,                     d            y                      d            x       of an implicit function, there is this formula,When   f  (  x  ,  y  )  =  0,                    d            y                      d            x        =  −            Differentiation of               f             w.r.t to               x             keeping               y             constant              Differentiation of               f             w.r.t to               y             keeping               x             constant      I want to understand how this method came about.Is there a clear proof for this formula?

broeifl · Accepted Answer

It&#039;s pretty easy to show if you use partial differentials with a slightly different notation. Let&#039;s let       ∂    x    f represent the partial differential of   f when we allow   x to freely vary (i.e., when   y is held constant). Note that when we allow   x to freely vary, then its differential is always the same as the total differential, so it can be notated as   d  x. Similarly with   y. So, the partial differential of   f with respect to   x is                     ∂        x            f              d      x      .Now, let&#039;s put this to use. A total derivative is the sum of its partials. Therefore, the total derivative of   f is going to be       ∂    x    f  +      ∂    y    f. Since the right-hand side is a constant, its derivative will be zero. Therefore:      ∂    x    f  +      ∂    y    f  =  0        ∂    x    f  =  −      ∂    y    f                      ∂        x            f                      ∂        y            f        =  −  1Now, what are our two partial derivatives? They are                     ∂        x            f              d      x       and                     ∂        y            f              d      y      . What happens if we put them in ratio with each other?                              ∂          x                f                    d        x                                      ∂          y                f                    d        y            Using fraction rules, we get (notice the &quot;-1&quot; comes from the equivalence above):                              ∂          x                f                    d        x                                      ∂          y                f                    d        y              =                    ∂        x            f              d      x        ⋅            d      y                      ∂        y            f        =                    ∂        x            f                      ∂        y            f        ⋅            d      y              d      x        =  −  1  ⋅            d      y              d      x        =  −            d      y              d      x      This is what we were looking for. On the left is the ratio of the two partial derivatives. On the right is the total derivative with a negative sign.

When f(x,y)=0, (dy)/(dx)=−(Differentiation of f w.r.t to x keeping y constant)/(Differentiation of f w.r.t to y keeping x constant)

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