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.