Certainly! Heres the derivation and calculation using standard formulas for an operational amplifier configured as a non$-$inverting amplifier, along with the specific values provided.### $($i$)$ Derive the Expression of the Voltage Gain and the Output VoltageFor a non$-$inverting operational amplifier, the voltage gain $\backslash ($ A$\_$v $\backslash )$ is given by the formula:$\backslash [$A$\_$v $=1+\backslash$frac$\{$R$\_2\}\{$R$\_1\}\backslash ]$Where:$-\backslash ($ R$\_1\backslash )$ is the resistor connected from the inverting input to ground.$-\backslash ($ R$\_2\backslash )$ is the feedback resistor connected from the output to the inverting input.Given:$-\backslash ($ R$\_1=4\backslash ,$ k$\backslash$Omega $=4000\backslash ,\backslash$Omega $\backslash )-\backslash ($ R$\_2=8\backslash ,$ k$\backslash$Omega $=8000\backslash ,\backslash$Omega $\backslash )$Substituting the values into the gain formula:$\backslash [$A$\_$v $=1+\backslash$frac$\{8000\backslash ,\backslash$Omega$\}\{4000\backslash ,\backslash$Omega$\}\backslash ]\backslash [$A$\_$v $=1+2=3\backslash ]$Thus$,$ the voltage gain $\backslash ($ A$\_$v $\backslash )$ is:$\backslash [$A$\_$v $=3\backslash ]$The output voltage $\backslash ($ V$\_\{$out$\}\backslash )$ can be calculated using the formula:$\backslash [$V$\_\{$out$\}=$ A$\_$v $\backslash$times V$\_\{$in$\}\backslash ]$### $($ii$)$ Determine the Values of the Voltage Gain and Output Voltage for the Input Voltage of $0\mathrm{.}3$VGiven:$-\backslash ($ V$\_\{$in$\}=-0\mathrm{.}3\backslash ,$ V $\backslash )$Now we can calculate the output voltage:$\backslash [$V$\_\{$out$\}=$ A$\_$v $\backslash$times V$\_\{$in$\}\backslash ]\backslash [$V$\_\{$out$\}=3\backslash$times $(-0\mathrm{.}3\backslash ,$ V$)\backslash ]\backslash [$V$\_\{$out$\}=-0\mathrm{.}9\backslash ,$ V$\backslash ]$### Summary of Results:$-**$Voltage Gain$**$: $\backslash [$ A$\_$v $=3\backslash ]-**$Output Voltage$**$ for $\backslash ($ V$\_\{$in$\}=-0\mathrm{.}3\backslash ,$ V $\backslash )$: $\backslash [$ V$\_\{$out$\}=-0\mathrm{.}9\backslash ,$ V $\backslash ]$