I am working on a problem in a textbook(Precalculus Mathematics for Calculus, by James Stewart) and the answer in the back of the book for the problem(1.4 #67) is -xy but I am not getting that. Here is the problem:
\begin{align} \frac{\frac{x}{y}-\frac{y}{x}}{\frac{1}{x^2}-\frac{1}{y^2}} \end{align}
Here is how I worked it out:
\begin{align} \frac{\frac{x}{y}-\frac{y}{x}}{\frac{1}{x^2}-\frac{1}{y^2}}=\frac{\frac{x^2-y^2}{xy}}{\frac{y^2-x^2}{x^2y^2}}=\frac{x^2-y^2}{xy}*\frac{x^2y^2}{y^2-x^2} =\frac{xy(x^2-y^2)}{y^2-x^2}=\frac{(xy)(x+y)(x-y)}{(y+x)(y-x)}=\frac{xy(x-y)}{y-x}=\frac{x^2y-xy^2}{y-x} \end{align}
Can someone please explain what I am doing wrong here?
You were almost there. $\frac{xy(x-y)}{y-x}$(your second last one)$=-xy$