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A derivation $D: R \to R $ is a additive homomorphism satisifying $$D(xy)=yDx + xDy$$ Let $K$ be the quotient field of $R$ , I want to show the derivation can be extended to $K$ such that the quotient law is well-defined $$ D(\frac{x}{y})=\frac{yDx-xDy}{y^2}$$

Here is what I got so far

Assume $\frac{x_1}{y_1}=\frac{x_2}{y_2}$, since R is an integral domain, one has $x_1y_2-x_2y_1=0$, so that $$ y_2Dx_1+x_1Dy_2-y_1Dx_2-x_2Dy_1=0$$ We need to show $$y_1y_2^2Dx_1-x_1y_2^2Dy_1-y_1^2y_2Dx_2+x_2y_1^2Dy_2=0$$ which is derived from the derivative of the quotients. I don't know how to connect those formulas. Any hint?

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1 Answer 1

up vote 4 down vote accepted

Hint: multiply the first formula by $y_1y_2$ and use $x_1y_2=x_2y_1$

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