For example: $$\begin{array}{l} y = {x^{11}}\\ \Delta x = 0.1 \cdot x\\ \Delta y =\sqrt {\left( {\frac{{dy}}{{dx}}\Delta x} \right)^2} = 1.1 \cdot {x^{10}}\\ y \pm \Delta y = {x^{10}} \pm 1.1 \cdot {x^{10}}\\ \pm \frac{{\Delta y}}{y} = \pm 110\% \end{array}$$
This above looks incorrect because more than 100%.
But the next $$\begin{array}{l} y = {x^{11}}\\ \Delta x = 0.1 \cdot x\\ x \pm \Delta x = \left( {1 \pm 0.1} \right) \cdot x \approx x \cdot {e^{ \pm 0.1}}\\ y \pm \Delta y = \left( {1 \pm {\delta _y}} \right) \cdot y \approx y \cdot {e^{ \pm {\delta _y}}} = {e^{\ln y \pm {\delta _y}}}\\ {\delta _y} = \frac{{d\ln y}}{{dx}}\Delta x = 1.1\\ y \pm \Delta y \approx y \cdot {e^{ \pm {\delta _y}}} = y \cdot {e^{ \pm 1.1}} = {x^{11}} \cdot {e^{ \pm 1.1}} = {\left( {x \cdot {e^{ \pm 0.1}}} \right)^{11}} \approx {\left( {\left( {1 \pm 0.1} \right) \cdot x} \right)^{11}}\\ \pm \frac{{\Delta y}}{y} ={e^{ \pm 1.1}} = \left[ \begin{array}{l} + 200\% \\ - 67\% \end{array} \right. \end{array}$$ looks acceptable.
Is there a proper name for such a method of error calculation? Or is such an approach unacceptable using logarithm?