$f\left( x \right)=\frac{{{\left( 2x-1 \right)}^{2}}}{2{{x}^{2}}}$
با استفاده از دستور ${{\left( \frac{u}{v} \right)}^{\prime }}=\frac{{u}'v-{v}'u}{{{v}^{2}}}$ داریم:
${f}'\left( x \right)=\frac{{{\left( {{\left( 2x-1 \right)}^{2}} \right)}^{\prime }}\left( 2{{x}^{2}} \right)-{{\left( 2{{x}^{2}} \right)}^{\prime }}{{\left( 2x-1 \right)}^{2}}}{{{\left( 2{{x}^{2}} \right)}^{2}}}$
${f}'\left( x \right)=\frac{2\left( 2 \right){{\left( 2x-1 \right)}^{1}}\times 2{{x}^{2}}-4x{{\left( 2x-1 \right)}^{2}}}{{{\left( 2{{x}^{2}} \right)}^{2}}}$
${f}'\left( x \right)=\frac{4x\left( 4{{x}^{2}}-2x-4{{x}^{2}}-1+4x \right)}{4{{x}^{4}}}=\frac{2x-1}{{{x}^{3}}}$