Using the Power Rule in Electrical Engineering

Ohm’s Law states that the current \(I\) running through a wire is inversely proportional to the resistance \(R\) in the wire and can be written as \(I=\dfrac{ V}{R}\), where \(V\) is the voltage. Find the rate of change of \(I\) with respect to \(R\) when \(V\)=12 volts.

Solution The rate of change of \(I\) with respect to \(R\) is the derivative \(\dfrac{dI}{dR}\). We write Ohm’s Law with \(V=12\) as \(I=\dfrac{V}{R }=12R^{-1}\) and use the Power Rule. \[ \dfrac{dI}{dR}=\dfrac{d}{dR}( 12R^{-1}) =12\cdot \dfrac{d}{dR} R^{-1}=12( -1R^{-2}) =-\dfrac{12}{R^{2}} \]

The minus sign in \(\dfrac{dI}{dR}\) indicates that the current \(I\) decreases as the resistance \(R\) in the wire increases.