deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 8b7fe10..cb8952f 100644  --- a/untitled.tex  +++ b/untitled.tex 
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And the second derivative with respect to $x$ is   \begin{align*}  (ln|f(x,y)|)_{xx}&= (\frac{uu_x+vv_x}{u^{2}+v^{2}})_x\\  &= \frac{(uu_x + uu_{xx} +v_xv_x + vv_{xx})(u^{2}+v^{2})-(uu_x + vv_x)(2uu_x+2vv_x)}{(u^{2}+v^{2})^{2}} \end{align*}  Similarly the first and second derivatives of $ln|f(x,y)|$ with respect to $y$ are  \begin{align*} 
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&=\frac{2uu_y+2vv_y}{2(u^{2}+v^{2})}\\  &=\frac{uu_y+vv_y}{u^{2}+v^{2}}  \end{align*}  and  \begin{align*}  (ln|f(x,y)|)_{yy}&= (\frac{uu_y+vv_y}{u^{2}+v^{2}})_y\\  &= \frac{(uu_y + uu_{yy} +v_yv_y + vv_{yy})(u^{2}+v^{2})-(uu_y + vv_y)(2uu_y+2vv_y)}{(u^{2}+v^{2})^{2}}  \end{align*}  \end{proof}